Ramsey Theory: Unsolved Problems and Results 9783110576702, 9783110576511

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Xiaodong Xu, Meilian Liang, Haipeng Luo Ramsey Theory

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Xiaodong Xu, Meilian Liang, Haipeng Luo

Ramsey Theory

| Unsolved Problems and Results

Mathematics Subject Classification 2010 Primary: 05C55, 05D10, 05C35; Secondary: 11B25, 68R10 Authors Xiaodong Xu Guangxi Academy of Sciences Nanning, 530007, P.R. China [email protected]

Haipeng Luo Guangxi Academy of Sciences Nanning, 530007, P.R. China [email protected]

Meilian Liang School of Mathematics and Information Science, Guangxi University, Nanning, 530004, P.R. China [email protected]

ISBN 978-3-11-057651-1 e-ISBN (PDF) 978-3-11-057670-2 e-ISBN (EPUB) 978-3-11-057663-4 Library of Congress Cataloging-in-Publication Data Names: Xu, Xiaodong, author. | Liang, Meilian, author. | Luo, Haipeng, author. Title: Ramsey theory : unsolved problems and results / Xiaodong Xu, Meilian Liang, Haipeng Luo. Description: Berlin ; Boston : De Gruyter, [2018] | Includes bibliographical references and index. Identifiers: LCCN 2018016483| ISBN 9783110576511 (print) | ISBN 9783110576634 (e-book (epub) | ISBN 9783110576702 (e-book (pdf) Subjects: LCSH: Ramsey theory. | Combinatorial analysis. | Graph theory. Classification: LCC QA166 .X865 2018 | DDC 511/.66–dc23 LC record available at https://lccn.loc.gov/2018016483 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 University of Science and Technology of China Press, Walter de Gruyter GmbH, Berlin/Boston Cover image: Can be found under the following link: https://commons.wikimedia.org/wiki/File: K_16_partitioned_into_three_Clebsch_graphs_twisted.svg This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International, 3.0 Unported, 2.5 Generic, 2.0 Generic and 1.0 Generic license. Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface To make combinatorics a more unified mathematical branch, we should use more unified methods to research more unified problems. In this book, we survey some conjectures and unsolved problems in Ramsey theory, and propose some new ones. We consider not only the problems and conjectures themselves, but also the relations between them. In fact, in this book, we focus on the relations between different problems, rather than choosing some problems that are believed to be more important, more famous, or more difficult among a lot of unsolved problems and conjectures in Ramsey theory. We believe that such a patten of research will be useful for those extremely difficult problems, in particular those with many variants. Most problems in this book can be partitioned into two classes of topics. One class is in extremal graph theory and the other in additive number theory. Most of the other problems are included in discrete geometry, functional analysis, algorithm design, complexity analysis, and so on. Of course, there need not be a clear line between the different branches of mathematics involved in this book. Some problems may involve enumerative combinatorics, analytic number theory, algebra, algebraic geometry, or other branches of mathematics. In some cases, a problem in this book may involve two or more branches in mathematics. Some problems in this book were cited from references and others were proposed by the authors; in most cases they were proposed by me and became what they are now through discussions among the authors. During those discussions, the understanding of some problems was deepened, and some problems were solved or proved to be trivial or disproved, and are not included in this book. Ramsey proved his famous theorem in 1928 when he was working on the decision problem for first-order predicate calculus with equality. About 90 years have passed, and Ramsey theory has developed into a large branch in combinatorics from a few theorems that seemed to be isolated from each other. The philosophy of Ramsey theory lies in that if a structure is large enough, then there must be a large substructure that is highly ordered. The research in Ramsey theory has led to many powerful methods. For instance, the study on the lower bound for the Ramsey number R(k, k) led to the probabilistic method, of which the influence is both wide and deep. Ramsey theory has promoted the intersection of combinatorics and other branches of mathematics. It has been used in many branches in mathematics, including some branches that were believed to be far from combinatorics, for instance, functional analysis. Ramsey’s theorem is a deep generalization of the Drawer principle, which is simple but commonly used. Therefore, it is not surprising that Ramsey theory has many applications. On the other hand, many mathematical tools in algebra, number theory, probabilistic theory, etc., have found many applications in combinatorics, including in Ramsey theory. Ramsey theory has important applications in commu-

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VI | Preface

nication theory, computer science, etc. In the last two decades, we have witnessed a tremendous flow of outstanding results in Ramsey theory. This book is suitable for professional mathematicians and computer scientists, as well as students, especially for those majoring in combinatorics or algorithm design and analysis. We would like the unsolved problems and conjectures in this book to become research topics for some graduate students and young mathematicians and computer scientists. We believe that even to obtain a partial solution to an interesting unsolved problem is better than to do too many dull exercises in many textbooks. Note that the authors have spent much time only on some problems in this book and may have overlooked the obvious in some cases. In a paper titled “Unsolved problems in number theory”, András Sárközy wrote the following words. (What we would like to say about this book is similar.) If there will be just one talented young mathematician who will write one of his first papers starting out from one of the problems below then this paper will achieve its purpose.

Although problems and related results in Ramsey theory in reference citations seem systematically limited to every topic, they are not systematic as a whole. Among the works in references, the differences between the importance of different works may be very large. The authors of some reference citations know very little about results that have a relation to the problems they are working on. Screening, integration, and further research on these problems are worthwhile doing. Many unsolved problems in Ramsey theory and related partial results are spread over the literature. Among these problems, some were discussed in the book titled Ramsey Theory written by Graham, Rothschild, and Spencer. Unsolved problems in Ramsey theory are important topics in Chung and Graham’s book Erdős on Graphs: His Legacy of Unsolved Problems. Many problems and results in Ramsey theory on integers can be found in the book Ramsey Theory on the Integers by Landman and Robertson. There are a few other books on Ramsey theory that include some unsolved problems and conjectures. However, there is no book especially on unsolved problems and related results in Ramsey theory that includes both problems in graph theory and problems in additive number theory. Therefore, we may say that this book is a new attempt in this area. We hope that such a book will be useful and interesting for readers. Richard Guy wrote a book on unsolved problems in number theory. To write a similar book on Ramsey theory, we carried out many discussions before beginning to write it in 2013. At the beginning we found that it was hard work to write a book of this type and later we found that it is even more difficult than we expected. The book covers many topics, and this coupled with the limited level of experience of the authors, and the fact that in-depth research was done only on some problems, errors and inappropriacies in the book are inevitable; there may be some simple prob-

Preface |

VII

lems that have been discussed as difficult ones. We hope that readers can put forward valuable opinions or exchange ideas and discuss with us. Research was partially supported by the National Natural Science Foundation (11361008) and the Guangxi Natural Science Foundation (2011GXNSFA018142). Xiaodong Xu Nanning March 2017

Contents Preface | V 1 1.1 1.2

Some definitions and notations | 1 Some definitions in graph theory | 1 Basic definitions in graph Ramsey theory | 3

2 2.1 2.2 2.3 2.4 2.5 2.6

Ramsey theory | 5 Unsolved problems and conjectures in Ramsey theory | 5 Frank P. Ramsey | 7 Paul Erdős and his work on Ramsey theory | 8 Some remarks | 10 Some interesting books and surveys | 11 The common thought in Ramsey theory | 12

3 3.1 3.2 3.3 3.4

Bi-color diagonal classical Ramsey numbers | 17 The differences between consecutive Ramsey numbers | 17 Some known results on diagonal Ramsey numbers | 19 Constructive lower bound for R(k, k) | 20 R(5, 5) and almost regular Ramsey graphs | 21

4 4.1 4.2 4.3 4.4

Paley graphs and lower bounds for R(k, k) | 25 Known lower bounds on small R(k, k) based on Paley graphs | 25 Some known results on the clique numbers of random graphs | 25 Some problems and conjectures on Paley graphs | 27 The lower bound for cl(G p ) based on the greedy algorithm | 28

5 5.1 5.2 5.3 5.4 5.5 5.6 5.6.1 5.6.2 5.6.3 5.7 5.8

Bi-color off-diagonal classical Ramsey numbers | 33 Some known bounds for off-diagonal Ramsey numbers | 33 Some problems and results on off-diagonal Ramsey numbers | 34 Lower bounds for some small off-diagonal Ramsey numbers | 37 Upper bounds for bi-color off-diagonal classical Ramsey numbers | 39 On R(3, s) and R(K3 , K s − e) | 39 Why are these problems on R(3, k) so difficult? | 41 Another idea | 43 What makes ∆ s < s difficult to prove? | 43 What makes ∆ s > 3 difficult to prove? | 43 When does R(l, s + t − 2) ≥ R(l, s) + R(l, t) − 1 holds | 45 Problems of algorithms on off-diagonal Ramsey numbers | 46

X | Contents

5.9 5.10 5.10.1 5.10.2 6 6.1 6.2 6.2.1 6.2.2

Constructive lower bounds for off-diagonal Ramsey numbers | 49 Ramsey graphs | 50 Disjoint k − 1-cliques in (k, l)-Ramsey graphs | 52 Turán type theorems for k-connected graphs | 52

6.2.3 6.3 6.4 6.5 6.6 6.6.1 6.6.2

Multicolor classical Ramsey numbers | 55 Results obtained by methods used on R(s, t) | 55 On the multicolor classical Ramsey number R n (k) | 56 The Shannon capacity of graphs | 56 R n (3) and the Shannon capacity of graphs with bounded independence numbers | 57 More problems on R n (k) | 61 Other multicolor classical Ramsey numbers | 64 Monotony conjecture on (k1 , k2 , . . . , k t ) graphs | 65 The Mathon–Shearer construction | 66 More remarks | 66 Remarks on constructive methods | 66 Remarks on computing | 67

7 7.1 7.2 7.3 7.4 7.5 7.5.1 7.5.2

Generalized Ramsey numbers | 69 Frank Harary and generalized Ramsey numbers | 69 A generalization of a known result on classical Ramsey numbers | 70 Some problems relative to R(C m , K n ) | 71 The connected Ramsey number | 73 The Ramsey–Turán problem | 74 The Turán problem with bounded chromatic numbers | 74 The Ramsey–Turán number | 74

8 8.1 8.2 8.2.1 8.2.2 8.2.3 8.2.4 8.2.5 8.3 8.3.1 8.3.2 8.4 8.5

Folkman numbers | 77 Some known results on Folkman numbers | 77 Small Folkman numbers | 79 On F e (3, 3; 4) | 79 A small example | 80 Bounds on F e (K4 − e, K4 − e; K4 ) | 81 Bounds on F v (4, 4; 5) | 82 More problems on small vertex Folkman numbers | 82 Bounds on some small generalized Folkman numbers | 83 On F v (K3 − e, K3 − e; K3 ) | 84 Bounds on F v (K4 − e, K4 − e; K4 ) | 85 On the generalized Ramsey numbers of Erdős and Rogers | 86 Other problems on Folkman numbers | 88

Contents | XI

8.6 8.7 8.8 9 9.1 9.2 9.3 9.4

F v (3, k; k + 1) and other vertex Folkman numbers | 90 Universal graphs | 93 On the connectivity of Folkman graphs | 94 The Erdős–Hajnal conjecture | 95 The Erdős–Hajnal conjecture and its multicolor generalization | 95 The Erdős–Hajnal number | 95 On the lower bound on the vertex Folkman number F v (k, k; k + 1) | 98 More generalizations | 99

10 Other Ramsey-type problems in graph theory | 101 10.1 Ramsey-type numbers based on directed graphs | 101 10.1.1 Directed Ramsey numbers for acyclic subtournaments | 101 10.1.2 Directed Ramsey numbers | 103 10.1.3 Directed Folkman numbers | 103 10.2 Hypergraph Ramsey numbers | 104 10.3 Size Ramsey numbers | 106 10.4 Ramsey multiplicities | 107 10.5 Induced Ramsey numbers | 107 10.6 The bipartite Ramsey number | 108 11 On van der Waerden numbers and Szemerédi’s theorem | 111 11.1 Known bounds for r k (n) | 112 11.2 The van der Waerden number W(k, k) | 113 11.3 Set-coloring generalization of van der Waerden numbers | 115 11.4 The difference between various van der Waerden numbers | 116 11.5 Upper and lower bounds for r p (p 2 ) | 117 11.6 More problems | 119 11.6.1 The Waerden–Szemerédi problem | 119 11.6.2 The Anti-Szemerédi problem | 120 11.6.3 A problem of Graham concerning monochromatic k-AP | 120 12 More problems of Ramsey type in additive number theory | 123 12.1 Problems related to the Hales–Jewett theorem | 123 12.2 Schur numbers and Rodo numbers | 124 12.3 Schur numbers | 125 12.3.1 A problem on sum-free subset of a given set | 126 12.4 Rado numbers | 127 12.4.1 A theorem of Croot on sums of unit fractions | 127 12.4.2 On x 2 + y 2 = z2 | 128 12.5 Rado numbers in group theory | 129

XII | Contents

13 13.1 13.2 13.3 13.4

Sidon–Ramsey numbers | 131 Basic definitions | 131 On small Sidon–Ramsey numbers | 132 Upper bounds for Sidon–Ramsey numbers | 133 On Golomb rectangles | 134

14 14.1 14.2 14.3

Games in Ramsey theory | 137 The Ramsey graph game | 137 Folkman games | 138 The van der Waerden game | 139

15 Local Ramsey theory | 141 15.1 Local Ramsey numbers | 141 15.1.1 More inequalities on R(K m , k − loc, t) | 142 15.1.2 Off-diagonal generalization of the local Ramsey number | 143 15.2 Local Folkman numbers | 144 15.3 Local van der Waerden numbers | 144 16 16.1 16.2 16.3 16.4 16.5

Set-coloring Ramsey theory | 147 Multigraph Ramsey numbers | 147 Multigraph Ramsey numbers and other Ramsey-type problems | 148 (2) Constructive lower bounds in Ramsey theory and f3 (q) | 149 Set-coloring Folkman numbers | 149 Some remarks | 151

17 17.1 17.2 17.3 17.4

Other problems and conjectures | 153 Rainbows and anti-Ramsey-type problems | 153 Banach spaces and Ramsey theory | 153 The Erdős–Szekeres theorem | 155 The chromatic number of the Euclidean plane | 157

Epilogue | 159 Bibliography | 163 Index | 177

1 Some definitions and notations In such a book of about 200 pages, many different topics in Ramsey theory or that are related to Ramsey theory will be discussed. We will use a large amount of terminology and notations. Hence, it is impossible for it to be self-contained. In this chapter, we list some basic definitions in graph theory and graph Ramsey theory. More often than not, in this chapter, we will define only such terminology and notations that will be frequently used in this book. We will not cite the definitions of Folkman numbers, van der Waerden numbers, etc. here; instead they will be defined in the subsequent chapters devoted to them. Throughout this book, let a, b, s, t, k, l, m, n, r and k i be positive integers. In many similar cases, we suppose that the numbers we use are positive integers, unless otherwise specified. The cardinality of a finite set A is denoted by |A|. Let [n] = {1, . . . , n} denote the set consisting of the first n positive integers. The set of all positive integers is denoted by ℕ, and the set of all integers is denoted by ℤ.

1.1 Some definitions in graph theory We will only write a small introduction to the terminology in graph theory that will be used in the book. Fortunately, much of the standard graph theoretic terminology is so intuitive that it is easy to understand. Some definitions in graph theory that cannot been found in this section, for instance, the Hamiltonian graph, may be found in the textbook [45] by Bondy and Murty, or the textbook [78] by Reinhard Diestel. Note that the textbook of Bondy and Murty that we suggest here is the 1976 version, not the advanced course of many more pages that was published in 2007. We believe that for graph theory, unlike for most of other branches of mathematics, an advanced course that includes many topics may not be necessary for most readers because they will be only interested in a few topic, and an advanced course would be used only as a handbook. However, as a handbook, some parts of these advanced courses on graph theory may soon be outdated. An advanced course on a few related topics in graph theory may be more useful. Suppose G is a graph. The set of its vertices is denoted by V(G), and the set of its edges by E(G). |V(G)| and |E(G)| are called the order and size of graph G, respectively. Sometimes we denote the order of graph G by n(G) or |V(G)|. For a finite graph G, both |V(G)| and |E(G)| are finite. All graphs considered in this book are finite graphs, unless otherwise specified. For graph G, the complement of G denoted by G is the graph with the vertex set V(G), and two different vertices in V(G) are adjacent in G if and only if they are not adjacent in G. The subgraph of G induced by S is denoted by G[S], where S ⊆ V(G).

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A graph that may contain multiple edges but does not contain loops is called a multigraph. In this book, we will not deal with graphs with loops. More often than not, we may call a graph without multiple edges “graph G” and call it “multigraph G” if it contains (or at least may contain) multiple edges. Let G1 = (V1 , E1 ) and G2 = (V2 , E2 ) be two graphs. We call G1 and G2 isomorphic, if there exists a bijection φ : V1 → V2 with uv ∈ E1 ⇔ φ(u)φ(v) ∈ E2 for all different vertices u, v ∈ V1 . Such a map φ is called an isomorphism. Let G1 ∪ G2 be graph H, where V(G1 ) ∩ V(G2 ) = 0, V(H) = V(G1 ) ∪ V(G2 ), and E(H) = E(G1 ) ∪ E(G2 ). Let G1 + G2 be graph H, where V(G1 ) ∩ V(G2 ) = 0, V(H) = V(G1 ) ∪ V(G2 ), and E(H) = E(G1 ) ∪ E(G2 ) ∪ {uv | u ∈ V(G1 ), v ∈ V(G2 )}. The composition of simple graphs G and H is the simple graph G[H] with vertex set V(G) × V(H), in which (u, v) is adjacent to (u 󸀠 , v󸀠 ) if and only if either uu 󸀠 ∈ E(G) or u = u 󸀠 and vv󸀠 ∈ E(H). G[H] is also called the lexicographic product of G and H; G[H] is important in some chapters in this book, in particular in Chapter 6 on multicolor classical Ramsey numbers. The clique number of G is the cardinality of the largest clique in G. Similarly, the independence number of G is the cardinality of the largest independent set in G. The clique number of G is denoted by cl(G), and the independence number of G is denoted by α(G). It is easy to see that α(G) = cl(G). Note that the independence number is called stability number in some references, and the clique number of G is denoted by ω(G) in some references. ∆(G) and δ(G) are the maximum and minimum degrees of G, respectively. If ∆(G) = δ(G) = d, then G is called a regular graph, and we also say that G is d-regular. The chromatic number of a graph G is the smallest number of colors needed to color the vertices in V(G) so that no two adjacent vertices share the same color. The chromatic number of G is denoted by χ(G). It is obvious that χ(G) ≥ cl(G) and α(G) ≥ |V(G)|/χ(G). Let N G (v) be the subset of V(G) including all neighbors of v, and N G− (v) be the subset of V(G) including all non-neighbors of v. A complete graph of order n is denoted by K n . A cycle of order n is denoted by C n , and a path of order n is denoted by P n . K3 is also called triangle. G is called kconnected (for k ∈ ℕ) if |G| > k and G − X is connected for every set X ⊆ V with |X| < k. The largest integer k such that G is k-connected is the connectivity κ(G) of G. The Turán graph T n,r , is the complete r-partite graph with n vertices whose partite sets differ in size at most 1. Therefore, T2t,2 is the complete bipartite graph K t,t , and T2t+1,2 is the complete bipartite graph K t,t+1. Let the number of edges in the Turán graph T n,r be t(n, r). In 1941, Turán proved his famous theorem, which tell us that if graph G is a K r+1 free graph of order n with most edges, then G is isomorphic to T n,r . This theorem is called Turán’s theorem. The forbidden subgraph problem involves the determination of the maximum number of edges that an n-vertex graph may have if it contains no isomorphic copy of a fixed graph H. This number is called the Turán number for H and is denoted by ex(n, H).

1.2 Basic definitions in graph Ramsey theory

| 3

1.2 Basic definitions in graph Ramsey theory An (s, t)-graph is a graph that contains neither a clique of order s nor an independent set of order t. We denote by R(s, t) the set of all (s, t)-graphs. An (s, t)-graph of order n is called an (s, t; n)-graph. We denote by R(s, t; n) the set of all (s, t; n)-graphs. The Ramsey number R(s, t) is defined to be the smallest number n for which R(s, t; n) is empty. An (s, t)-graph of order R(s, t) − 1 is called an (s, t)-Ramsey graph. Note that differently to some authors, we do not call an (s, t)-graph of order smaller than R(s, t) − 1 an (s, t)-Ramsey graph. We can define an (s, t)-graph in another form as follows. If G is the complete graph K n of which all edges are colored with color 1 and color 2, and it contains neither a complete subgraph of order s of which all edges are in color 1 nor a complete subgraph of order t of which all edges are in color 2, then we call G an (s, t)-graph. For connected graphs G1 , . . . , G r , the multicolor Ramsey number R(G1 , . . . , G r ) is the smallest positive integer n, such that if we color the edges in K n with color 1, . . . , color r, then there is monochromatic G i in color i for some i ∈ [r]. Note that some other Ramsey-type numbers are called generalized Ramsey numbers in some references. In this book, we use the generalized Ramsey number only in the way in the definition above. We can define R(G1 , . . . , G r ) and the (G1 , . . . , G r )-Ramsey graph for connected graphs G1 , . . . , G r similarly. Now let us define arrow as follows. For a graph G and positive integers a1 , . . . , a r , if every r-coloring of the vertices in G must result in a monochromatic a i -clique of color i for some i ∈ [r], then we write G → (a1 , . . . , a r )v . We can define G → (H1 , . . . , H r )v for given graphs G, H1 , . . . , H r similarly. Similarly, for a graph G and positive integers a1 , . . . , a r , if every r-coloring of the edges in G must result in a monochromatic K a i of color i for some i ∈ [r], then we write G → (a1 , . . . , a r )e . It is not difficult to define the Ramsey numbers based on edge arrowing in another way, as follows. R(a1 , . . . , a r ) is the smallest positive integer n such that K n → (a1 , . . . , a r )e . Given an integer n ≥ 3, suppose that S ⊆ {1, 2, . . . , ⌊n/2⌋}. Let G be a graph with the vertex set V(G) = [n] and the edge set E(G) = {(x, y) | min{|x − y|, n − |x − y|} ∈ S}, and G is called a cyclic graph of order n, denoted by G n (S); S is called the parameter set of G n (S). Similarly, a cyclic (k 1 , . . . , k r ; n)-coloring of K n over ℤ n will be represented by a partition {C i }ri=1 of [n − 1] with the property that j ∈ C i implies n − j ∈ C i for all i ∈ [r] and j ∈ [n − 1]. The color of the edge (j1 , j2 ), 0 ≤ j1 < j2 < n, is equal to i if and only if j2 − j1 ∈ C i .

2 Ramsey theory The Ramsey theorem is a generalization of the Drawer principle. Contemporary research related to Ramsey theory spans many diverse areas of mathematics. In this chapter, we will not give a brief history of Ramsey theory by listing some important theorems in Ramsey theory. Instead, we will discuss unsolved problems and conjectures in Ramsey theory. Let us cite some words from an article by Graham on what Ramsey theory typically deals with. Graham wrote that Ramsey theory typically deals with problems of the following type. We are given a set S, a family F of subsets of S, and a positive integer r. We would like to decide whether or not for every partition of S = C1 ∪ ⋅ ⋅ ⋅ ∪ C r into r subsets, it is always true that some C i contains some F ∈ F. We will discuss Frank P. Ramsey and Paul Erdős and some of their work, respectively, which may be interesting for the reader. We will also list some interesting books and surveys, and discuss the common thought in Ramsey theory. However, before discussing them, let us start with remarking on the combinatorics of today. Combinatorics may be regarded as a mathematical branch similar to number theory in a way, because both include many unsolved problems that are easy to understand. Of course, as a mathematical branch combinatorics is much more discrete than number theory today. Different from many famous conjectures in number theory, most conjectures in combinatorics are known only by experts. It is difficult to predict when combinatorics will become mainstream mathematical branch in the future. Although there is a strong need from the point of view of computer science, we still believe that combinatorics is currently not mainstream for most mathematicians. Some mathematicians were awarded the Fields medal, the Wolf prize, or the Abel prize for their great works in combinatorics, and some works of these mathematicians are included in Ramsey theory or are related to Ramsey theory. However, this fact is not enough for us to believe that combinatorics is mainstream in the eyes of most mathematicians. Furthermore, who and which work can be awarded those famous prizes may be accidental in some cases. For instance, if Gowers had not obtained his new proof of Szemerédi’s theorem, and Green and Tao had not proved their famous theorem on long arithmetic progressions in primes, would Szemerédi be awarded the Abel Prize? It is difficult to answer, although Erdős believed that Szemerédi should have been awarded a Fields medal long ago.

2.1 Unsolved problems and conjectures in Ramsey theory In general, it is not difficult to propose a new conjecture in mathematics that we can neither prove nor disprove, or a problem that we cannot answer. It may be much easier https://doi.org/10.1515/9783110576702-002

6 | 2 Ramsey theory

to do so now with the help of the free online encyclopedia Wikipedia or other resources and tools than in 1900 when Hilbert presented his famous unsolved problems in mathematics. Therefore, it is not surprising that there are many difficult unsolved problems in mathematics, including many that have been proposed in the past few decades, for which people may have done much research and obtained some progression, but the present situation is far from solving the problems completely. In some cases, we have understood the problems deeply, but in many other cases we may be almost ignorant. Because mathematics is much larger now than it was in 1900, any list of unsolved problems in mathematics that is not very large is by no means exhaustive. Even an exhaustive list of unsolved important problems in combinatorics is not easy to write. On the other hand, if someone gives a list of problems, there need not be many people interesting in these problems, if the list is neither given by a famous mathematician nor with a big prize. Even the list of problems given at the turn of the century by Smale, a great mathematician of our time, can only be interesting for a few mathematicians. Nevertheless, we hope that this book will arouse the interest of mathematicians working in more mathematical branches. In particular, it is often not difficult to propose some new difficult problems in Ramsey theory. In many new papers on Ramsey theory, either new Ramsey-type problems are proposed and studied, or recently proposed Ramsey-type problems are studied. Most of these problems are generalizations or variants of related old problems. It is a pity that in many cases these new problems may be not as basic and interesting as the old ones, and working on these new problems seems not to be a valid way to understand the old problems better. Many new problems in Ramsey theory may be difficult, and the mathematicians who proposed them can only solve some simple cases in their papers and leave the problems isolated, or relate them to a well-known hard problem and then stop. In many mathematical papers, some conjectures were proposed only because the authors did not know and did want to know how to prove them, but did no necessary analysis and computation patiently. Even for some feasible but difficult topics, many authors do not have enough patience and bravery to translate the difficulty of quality into the difficulty of quantity and do little detailed analysis of many local cases that may be feasible. Such an attitude makes it nearly impossible to make progress on difficult problems. In some cases, studying the feasible subcases may help us to understand the entire problems better. We have to say that many people cannot do hard work and do not like to do easy work. On the other hand, sometimes a problem or conjecture may have a profound and lasting influence on a branch in mathematics. For instance, some conjectures on local zeta-functions derived from counting the number of points on algebraic varieties over finite fields, which were proposed by Weil in 1948, had great influence on algebraic geometry in the second half of the twentieth century. An older conjecture, the Riemann hypothesis proposed by Riemann in 1859, is believed to be the most important conjecture in analytic number theory. Many mathematicians believe that the Riemann

2.2 Frank P. Ramsey |

7

hypothesis is the most important conjecture in mathematics. It is not difficult to find some influential problems in Ramsey theory. For instance, the Erdős–Turán conjecture on arithmetic progressions that later became Szemerédi’s theorem. We cite the following two paragraphs by M. Kline from the preface of his masterpiece Mathematical Thought from Ancient to Modern Times. The usual courses in mathematics are also deceptive in a basic respect. They give an organized logical presentation which leaves the impression that mathematicians go from theorem to theorem almost naturally, that mathematicians can master any difficulty, and that the subjects are completely thrashed out and settled. The succession of theorems overwhelms the student, especially if he is just learning the subject. The history, by contrast, teaches us that the development of a subject is made bit by bit with results coming from various directions. We learn, too, that often decades and even hundreds of years of effort were required before significant steps could be made. In place of the impression that the subjects are completely thrashed out one finds that what is attained is often but a start, that many gaps have to be filled, or that the really important extensions remain to be created.

What M. Kline told us is correct and some young students may be taught based on some deceptive material. This is a book on unsolved problems in Ramsey theory. We believe that we should relate our topics to more interesting problems and give the readers more useful information. However, in such a book, it is not a good idea to cite the reference whenever we cite a sentence, because if we do so, then either the book will contain too many references or a footnote war may occur. Furthermore, we often make some remarks in the book. As we know, there are not enough useful remarks on mathematical work today, in particular negative ones.

2.2 Frank P. Ramsey Frank P. Ramsey was born on 22 February, 1903, and died on 19 January, 1930. He was a precocious British philosopher, mathematician, and economist. Ramsey proved the following theorem in [234]. Theorem 2.2.1 (Ramsey’s theorem). For every choice of positive integers p, k, n there exists an integer N with the following property: For every set X of a size at least N and for every partition A1 ∪ ⋅ ⋅ ⋅ ∪ A k of ( Xp) set of all p-subsets of X there exists a homogeneous subset Y of X of size at least n. Here, homogeneous means that (Yp ) is a subset of one of the classes of the partition. As D. M. Mellor wrote in [197], Ramsey solved a special case of the decision problem for first-order predicate calculus with equality. The irony is that, although Ramsey produced his theorem to help solve this problem, it can be solved without it. Moreover, Ramsey only solved this special case as a contribution towards

8 | 2 Ramsey theory

solving the general decision problem, an object which Gödel in effect showed to be unattainable the year after Ramsey died. So Ramsey’s enduring fame in mathematics rests on a theorem he didn’t need, proved in the course of trying to do something we now know can’t be done!

Mellor’s description is dramatic and true. On the other hand, we can remark on Ramsey’s paper as follows. It is obvious that our remark is completely different from that of Mellor. Let us cite what Ramsey wrote at the beginning of his paper [234]. This paper is primarily concerned with a special case of one of the leading problems of mathematical logic, the problem of finding a regular procedure to determine the truth or falsity of any given logical formula. But in the course of this investigation it is necessary to use certain theorems on combinations which have an independent interest and are most conveniently set out by themselves beforehand.

Although Ramsey proved the existence of Ramsey numbers and knew that his upper bounds are weak, he was not interested in improving them. However, he believed that his theorems are interesting. Gödel’s incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic. These results, published by Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert’s program to find a complete and consistent set of axioms for all mathematics is impossible. Gödel’s incompleteness theorems are negative theorems. Ramsey did not know what would happen when he wrote his paper [234]. Even if he knew, the aim in his paper is still reasonable, although not so important as before, because even if the general decision problem is know to be unattainable, it is still necessary to solve a special case. In [235] and [236] Ramsey studied taxation and saving, respectively. These works made Ramsey an important economist and were praised highly by Keynes. Based on what we can find in Wikipedia, we can see that Ramsey was the nominal supervisor of Wittgenstein, who had been a student of Cambridge before World War I and submitted his Tractatus Logico-Philosophicus as his doctoral thesis much later.

2.3 Paul Erdős and his work on Ramsey theory Paul Erdős was born in 1913 and died in September 1996 at the age of 83. He was well known for his fondness for travel and he lectured in more universities than anyone else. Erdős usually wrote a joint paper with one or more of the mathematicians at each university he visited. Erdős wrote many papers, most of which were co-authored with others. Differently from some Erdős fans, Erdős believed that we should not count papers, instead we should weigh them. Today, many people count papers rather than

2.3 Paul Erdős and his work on Ramsey theory

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weigh them; it is a serious problem not only for mathematicians but also for other scientists. Erdős researched many branches of mathematics and did important work in number theory and combinatorics. He won the Wolf prize in 1984, where his contribution is acclaimed as “for his numerous contributions to number theory, combinatorics, probability, set theory, and mathematical analysis, and for personally stimulating mathematicians the world over.” Many mathematicians that study combinatorics have been influenced by Erdős. Erdős did a lot of work in Ramsey theory, including proposing problems and conjectures and proving theorems. Just as written in the tribute given by Joel Spencer at the National Meeting of the American Mathematical Society in January 1997, Erdős’ place in the mathematical pantheon will be a matter of strong debate for in that rarefied atmosphere he had a unique style. As mentioned above, he was awarded the Wolf prize in 1984. Some people may believe that Erdős is not among the top 100 mathematicians of the twentieth century, but some people believe that he is one of the most great mathematicians of the twentieth century. In an article on the elementary proof of the Prime number theorem, Goldfeld pointed out that it is clear that Erdős founded a unique school of mathematical research, international in scope and highly visible to the world at large. Ernst Straus, who worked as an assistant to Albert Einstein long ago, was a German-American mathematician who helped found the theories of Euclidean Ramsey theory and of the arithmetic properties of analytic functions. Straus wrote a commemoration of Erdős’ 70th birthday. In the commemoration, Straus pointed out that in our century, in which mathematics is so strongly dominated by “theory constructors”, Erdős remained the prince of problem solvers and the absolute monarch of problem posers. Straus believed that Erdős was the Euler of our times, because the methods and results of Erdős’ work already let us see the outline of great new disciplines, such as combinatorial and probabilistic number theory, combinatorial geometry, probabilistic and transfinite combinatorics and graph theory, as well as many more yet to arise from his ideas. Erdős was without argument a master of the art of elementary methods. He showed the probabilistic method when he studied the lower bound for R(k, k) in [96], what is important in the history of combinatorics. Let us cite the following paragraph from an article titled “To prove and conjecture: Paul Erdős and his mathematics” written by Bollobás. Erdős’ ground-breaking work on random graphs with Alfred Renyi started in the late fifties: in a series of brilliant papers they laid the foundation of the theory of random graphs. The main discovery was that, for many a monotone increasing property, there is a sharp threshold: graphs of order n with slightly fewer edges than a certain function f(n) are very unlikely to have the property, while graphs with slightly more than f(n) edges are almost certain to have the property.

10 | 2 Ramsey theory

The probabilistic method has many applications and important influences. Let us cite some words of Noga Alon as follows. It is worth noting that although his argument seems trivial today, it was far from being obvious when published in 1947. In fact, several prominent researchers believed, before the publication of this short paper, that R(K k , K k ) may well be bounded by a polynomial in k.

Remark: [96] is, of course, a masterpiece, but in the 1940s there may only have been very few mathematicians who spent much time on studying the bounds for R(k, k). Turán conjectured that the Turán graph may be a good construction for the lower bound for R(k, k). That is, Turán conjectured that R(k, k) is bounded by a polynomial in k of degree 2. It is not surprising because Turán was the mathematician who proved Turán’s theorem, and it is not difficult to know what Turán conjectured was not true based on a construction of the form G[H], where G is isomorphic to C5 and H is a (k + 1, k + 1)-Ramsey graph, by what we have R(2k + 1, 2k + 1) − 1 ≥ 5(R(k + 1, k + 1) − 1) . Such work was done by Abbott later in the 1960s and looks easy today. Furthermore, we know that R4 (k)− 1 ≥ (R(k, k)− 1)2 , which was known by Abbott in [1] no later than in 1965. Although we can only improve this lower bound for R4(k)−1 a little, we believe that no mathematician will conjecture that the difference between the two sides of this inequality is very small for general k. The Ramsey theory that we see today includes the contributions of both mathematicians who obtained important results, and other mathematicians whose work has not seemed important until now. Let us refer to the “forerunners’ defect – deliberately exaggerated their new discovery” mentioned in Popper’s book titled Lessons of the 20th Century. When evaluating important mathematicians in history, some people also make similar mistakes. They almost infinitely exaggerate the importance of some pioneering work, including some work done by themselves. However, in our opinion, the following viewpoint of M. Kline’s in the preface to his Mathematical Thought from Ancient to Modern Times may be more reasonable. Although Newton and Leibniz are eminent predecessor, they even not understand many of the concepts of calculus thoroughly. Mathematicians cost about 200 years’ effort to get these concepts clearly.

We may utter similar words for Erdős and his great work on the probabilistic method.

2.4 Some remarks Although the history of Ramsey theory is nearly 90 years old, what we understand is still very limited. This needs more mathematicians’ effort in the future.

2.5 Some interesting books and surveys

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11

In the last 20 years, we have witnessed a tremendous flow of outstanding results in Ramsey theory, and more people have obtained more results based on the work of others. We would like to say, if someone can improve the basic inequality R(s, t) ≤ R(s − 1, t) + R(s, t − 1) and its multicolor generalization in general, or improve the method used by Erdős in studying the lower bound for R(k, k) in 1947, it may be more important work. In Ramsey theory, there had been some topics that were very hot for some time, but the new papers dealing with more difficult cases based on more sophisticated methods, may be difficult to be published later. For instance, papers on lower bounds for small Ramsey numbers, and papers on generalized Ramsey numbers that include only almost trivial results.

2.5 Some interesting books and surveys It is not difficult to find many works on Ramsey theory, and it is necessary to introduce some interesting books and surveys to the reader. There are some interesting books on open problems in mathematics. One of them is Arnold’s Problems [17]. For a much earlier book, we know of Hilbert’s Problems, which is very famous. Some mathematicians believe that Hilbert’s problems have had great influence on the development of mathematics in twentieth century. In the following part of this section, we will only consider books and surveys related to Ramsey theory. [147] is an important book on Ramsey theory written by R. Graham, B. Rothschild, and J. H. Spencer. The third edition was published in 2015. Mathematics of Ramsey Theory is a book edited by J. Nešetřil and V. Rödl in 1990 [209]. Some leading mathematicians are among the contributors. Erdős proposed many problems in combinatorics and number theory. Some problems proposed by Erdős on graphs in Ramsey theory can be found in [64], a book titled Erdős on Graphs – His Legacy of Unsolved Problems written by Chung and Graham, one of the most interesting books related to this book. Erdős wrote some problem papers, and some of these include problems in Ramsey theory. Many problems and results in Ramsey theory on the integers, can be found in [176] by B. M. Landman and A. Robertson. Small Ramsey numbers [229] written by S. P. Radziszowski, with the 14th version updated in 2014, is a standard reference on the subject published in the Electronic Journal of Combinatorics as a dynamic survey since 1994. [72] by Conlon, Fox, and Sudakov is a useful survey on recent developments in graph Ramsey theory. Many problems and theorems in Ramsey theory are on partitions, and we may regard them as colorings. Thus, the reader of this book may think that [270], The Mathematical Coloring Book written by Soifer, is interesting. Ramsey Theory: Yesterday, To-

12 | 2 Ramsey theory

day, and Tomorrow [271], edited by Soifer, includes some interesting articles. Unsolved Problems in Number Theory [152], written by Richard Guy who was 100 years old in 2016, is a famous book on the topic in its title. We often read some sections in [152], in which some problems of Ramsey-type were surveyed, including some on van der Waerden numbers, Szemerédi’s theorem, and Schur numbers.

2.6 The common thought in Ramsey theory People studying different problems in Ramsey theory may have different views on what problems in Ramsey theory are important. In the book titled The Best of All Possible Worlds written by Ivar Ekeland, we can find the following two sentences. The first is the verses that Voltaire wrote for Maupertuis: You have gone to confirm, in places far and lonesome What Newton always knew without leaving his disk.

The second is, As Mach puts it, Science itself can be considered as a minimum problem, consisting in accounting for facts as perfectly as possible, at the smallest intellectual expense.

For the first thought, maybe some mathematicians researching asymptotic bounds on Ramsey numbers can recall similar verses to people studying bounds on small Ramsey numbers with the help of computers. It is a pity that the case is different from that of Maupertuis and Newton, because for Ramsey theory, today it is impossible to find a giant matching Newton. Note that Newton established a great theory system, but in Ramsey theory, what we have done is only the first step in the long match. In an interview, Arnold discussed the differences in the way people from different cultures do mathematics. He believed that the International Science Foundation would do better to support mathematicians working in the good Russian style, which is to sit at home working hard to prove fundamental theorems that will remain the cornerstones of mathematics forever, rather than working in an American style, that is, traveling a lot to present all the latest results at all related conferences and being personally known to all experts in the field. For the second thought, what Mach discussed is the historical tradition of introducing as few concepts as possible to solve problems. More precisely, we should introduce as few and simple concepts as possible, not only consider the number of concepts, but also consider if it is economically based on the degree of acceptability of the concepts introduced. If we contrast the reality and the history of mathematics, perhaps we can find that the following conclusion is justified: at present, many mathematicians are almost do-

2.6 The common thought in Ramsey theory

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ing the opposite of this historical tradition. That is, introducing lots of new concepts and after having used them to study old and important problems and getting nowhere, then proposing new problems about these new concepts and studying them, and becoming narcissistic about many so-called innovations. However, more often than not, these new proposed problems are not important in themselves and cannot be used to better understand the old important problems. There are so many problems in Ramsey theory and it is easy and not necessary to find an example of this type in Ramsey theory. The purpose of the study of mathematical history is the thought behind the problems and results, which are among the most important purposes of the study of mathematics. Now, let us discuss common thought in Ramsey theory. Different from many conjectures in number theory, many conjectures in combinatorics are not only qualitative, but also quantitative. For instance, we do not know if the limit limk→∞ R(k, k)1/k exists; even if we suppose it exists, we do not know what the limit is and know only that it is between 21/2 and 4. The bounds for some Ramsey numbers obtained by a method may be the best known ones for parameters in some range, but are weaker than the bounds obtained by other methods for other parameters. For instance, we use different methods to study the asymptotic bounds on R(k, k) and the bounds on R(5, 5). We may say that they are two different things. Many mathematicians working on Ramsey theory may believe that it is more important to determine the asymptotic order of R(k, k) than to determine the exact value of R(5, 5). We believe that such a viewpoint is objective. As we know, in the long history of mathematics, there are some difficult problems that have not been considered as important until an important improvement appeared. We do not know if R(5, 5) will be a problem of this type. It seems that there is not enough evidence to prove that R(5, 5) cannot lead to important methods. More often than not, it is difficult and sometimes impossible to predict what will happen in the development of mathematics or to what method will a problem lead. Although the part on arithmetic progressions in this book has some relation to analytic number theory, the authors know only a little about analytic number theory. Even so, we like to cite a viewpoint of Hongquan Liu, written in the preface of his book on exponential sums, i.e., “Many people look down on the small improvements on some data in analytic number theory, in contrast, I believe that it is very important . . . ”. We believe that Liu’s viewpoint is correct, and for small improvements on some asymptotic bounds in combinatorics we may discuss similarly. When we contrast the bounds obtained by different methods, sometimes we need exact bounds rather than those including o(1) or ϵ. Some asymptotic results are not exact, that is to say, to know what the bound of the function is in the conclusion when some parameters equal to given values, simply substitutions of the values of the related parameters is not enough, and we must go back to the details of the process of getting these results in order to do the related analysis and computing once more specially for this aim. In most cases, the study of the exact forms of those known

14 | 2 Ramsey theory

non-exact results, may be regarded as exercises for beginners, but in some cases this work may be not easy. The versions in exact forms may be useful for people who want to compare the results obtained by one method with those obtained by other methods, in particular to refer to when evaluating the algorithms and results on small cases. Mathematicians may be guided by small improvements to search for new important methods. However, these small improvements have not been regarded as improvements by some mathematicians studying only asymptotic bounds. Most small improvements cannot become great ones, but if one is always thinking about great breakthroughs without thinking of any small improvements in detail, then it may be difficult to obtain any new results on difficult problems. More often than not we should cite the bounds in detail with most information that may be useful for the reader. Ramsey theory may be more general than some people believe. For instance, Turán’s theorem is on the problem of how many edges a K k -free graph of order n can have. Different from Ramsey theory, the complement of G is not considered directly. More often than not we will not classify Turán’s theorem as a Ramsey-type theorem. However, we can consider this problem in another way. The following form of Turán’s theorem may be regarded as a Ramsey-type theorem. Suppose that n and k are integers and n ≥ k ≥ 2. For any graph G of order n, either G contains K k , or its complement graph has at least C2n − t(n, k − 1) edges. Let us now discuss enumeration and Ramsey numbers. If we know the number of K s -free graphs of order n, the graphs with independence numbers smaller than t, and the graphs that have both cliques of order s and independent sets of order t, then we can determine if there is an (s, t)-graph of order n. Therefore, we can determine the exact value of R(s, t) if we know these for integers s, t, and n, including the case s = t = 5 and n = 43. Now, we can do none of them. As a unified subject, Ramsey theory has a common thought behind those different problems in it. However, these problems are on different topics, and we need different methods to research them. How to understand the common thought in Ramsey theory more deeply by making large improvements on concrete important problems is a longterm task for mathematicians working on Ramsey theory. There are many problems and conjectures in Ramsey theory. Frank Harary and some co-authors wrote a series of papers on generalized Ramsey numbers. In [158], Frank Harary told the story of these papers. He said that most of the time with independent discoveries, he was the winner, because he fell in love with graph theory in 1950, which was earlier than most people who were then (1980) fascinated by graphs. Now, it is not easy to propose new problems that are both natural and interesting, not like was the case dozens of years earlier. We can see that among those problems that were studied by people, there are some interesting ones, but many among them are not so interesting. How can we use Ramsey theory in designing better algorithms? This is a vague question, but we may find its importance in the future.

2.6 The common thought in Ramsey theory

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The following Chapters 3, 4, and 5 are on bi-color classical Ramsey numbers. Among them, bi-color diagonal classical Ramsey numbers are discussed in Chapter 3, Paley graphs and lower bounds for R(k, k) are discussed in Chapter 4. All of them are on diagonal classical Ramsey numbers. Bi-color off-diagonal classical Ramsey numbers are discussed in Chapter 5.

3 Bi-color diagonal classical Ramsey numbers The bi-color diagonal classical Ramsey number R(k, k) is one of the most important and interesting topics in Ramsey theory. We will list some conjectures and problems on R(k, k). Let us discuss the differences between consecutive Ramsey numbers in the first section.

3.1 The differences between consecutive Ramsey numbers We know that R(k, s + t − 1) ≥ R(k, s) + R(k, t) − 1 , which can be proved based on a simple construction G1 ∪ G2 , where G1 is a (k, s)Ramsey graph and G2 is a (k, t)-Ramsey graph. This inequality is simple. We find a proof given by Li Weizheng in 1982, written in Chinese. Maybe it is the earliest one. The following two theorems were proved by constructive methods as parts of Theorems 2 and 3 in [313], and Theorem 3.1.2 is a corollary of Theorem 3.1.1. Theorem 3.1.1 ([313]). Given a (k, s)-graph G and a (k, t)-graph H, for some k ≥ 3 and s, t ≥ 2, if both G and H contain an induced subgraph isomorphic to some K k−1 -free graph M, then R(k, s + t − 1) ≥ n(G) + n(H) + n(M) + 1 . Theorem 3.1.2 ([313]). If 2 ≤ s ≤ t and k ≥ 3, then {k − 3 , R(k, s + t − 1) ≥ R(k, s) + R(k, t) + { k−2, {

if s = 2 ; if s ≥ 3 .

We can see that in Theorem 3.1.1 if G is (k, s)-Ramsey graph and H is a (k, t)-Ramsey graph, then we can obtain a lower bound for R(k, s + t − 1) that is better than R(k, s) + R(k, t) − 1. In [313], Theorem 3.1.1 was used to obtain lower bounds for some small Ramsey numbers of the form R(k, s + t − 1), where k ≥ 4. When k = 3, we cannot improve those best known lower bounds for small Ramsey number R(3, s) by Theorem 3.1.1. Later, the case k = 3 of Theorem 3.1.1 was used to study the chromatic gap and (3, s)-Ramsey graphs with large chromatic numbers by A. Gyárfás, A. Sebõ, and N. Trotignon in [155], and it is also a main tool in [33], a paper on large chromatic numbers and Ramsey graphs by Biró, Füredi, and Jahanbekam. The first inequality of Theorem 3.1.2 for s = 2, R(k, t + 1) ≥ R(k, t) + 2k − 3 , was proved by Burr, Erdős, Faudree, and Schelp in 1989 [54]. This lower bound was improved by 1 for k ≥ 5 in [309]. Note that in [167], a paper published in 1967, Kalbfleisch https://doi.org/10.1515/9783110576702-003

18 | 3 Bi-color diagonal classical Ramsey numbers

wrote that “It appears that a result of the form R(q, n) ≥ R(q, n − 1) + 2q − 4 might hold under suitable restrictions on n, q.” In 1980, Paul Erdős wrote in [98], page 11 (using r for our R): Faudree, Schelp, Rousseau, and I needed recently a lemma stating lim

n→∞

r(n + 1, n) − r(n, n) =∞ n

(a)

We could prove (a) without much difficulty, but could not prove that r(n + 1, n) − r(n, n) increases faster than any polynomial of n. We of course expect lim

n→∞

1 r(n + 1, n) = C2 , r(n, n)

(b)

where C = limn→∞ r(n, n)1/n .

Based on work reported in [309], the best known lower bound estimate for the difference in (a) seems to be barely Ω(n). Between 2007 and 2009 Radziszowski and Xiaodong Xu asked several mathematicians about this, including Erdős’ co-workers mentioned in the article [98] cited above. Nobody could recall the proof or even its existence. The conclusion Radziszowski and Xiaodong Xu were inclined to draw in [309] is that there exists no known proof of (a), although possibly it was known to Erdős. It is thus prudent to consider (a) at the moment to be only a conjecture. Now, let us guess the possible method by which Erdős proved (a), if he proved it. Let us consider the following problem. Problem 3.1.1. Is there a positive integer k 0 such that R(k, k + 1) + c ≥ R(k, k) + R(3, k) for any integer k ≥ k 0 ? Here, c is a positive constant. We can see that if the answer to this problem is yes, then (a) holds. However, (a) need not have been proved by Erdős, let alone proved by solving Problem 3.1.1. Note that in 1980, the best known lower bound for R(3, k) was ck 2 /(log2 k). Let G1 be a (k, k)-Ramsey graph and G2 be a (k, 3)-Ramsey graph, where V(G1 ) ∩ V(G2 ) = 0. We do not know if we can construct a (k, k + 1)-graph G by adding edges between V(G1 ) and V(G2 ) in G1 ∪ G2 , even for k large enough. If we can obtain a graph G that is not a (k, k + 1)-graph, but G contains an induced subgraph that is a (k, k + 1)graph obtained by deleting no more than c1 k vertices for some positive constant c1 , then we can prove (a). This is far from being reached now. We know very few related results. As mentioned earlier, in [54], R(s, t + 1) ≥ R(s, t) + 2s − 3

3.2 Some known results on diagonal Ramsey numbers

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was proved, and in [309] more results were proved, including R(s, t+1) ≥ R(s, t)+2s−2 for s ≥ 5, which was proved based on the following Theorem 3.1.3. We do not know if there is a positive integer k0 such that R(k, k + 1) + c ≥ R(k, k) + 3k for any k ≥ k 0 , and we may try to prove it first. In [309], the following theorem was proved. Theorem 3.1.3. Let k ≥ 4 and s, t ≥ 2. Given a (k, s)-graph G and a (k, t)-graph H, let M be a graph isomorphic to induced subgraphs of G and H. If the vertex set VM can be partitioned into two nonempty sets, VM = W1 ∪ W2 , so that for every v ∈ VM − W i and i ∈ {1, 2} there is no K k−1 in M[W i ∪ {v}], then R(k, s + t − 1) ≥ n(G) + n(H) + n(M) + 1 . Now, let us consider the following problem and conjecture on R(k, k) and R(k, k + 1). Problem 3.1.2. Does (R(k, k + 1))/(R(k, k)) > 1 + c hold for some positive constant c? Maybe it is easier to prove the following conjecture than to answer Problem 3.1.2. Conjecture 3.1.1. For any given positive integer s, lim

k→∞

R(k, k + 1) − R(k, k) =∞. ks

Let s be any given positive integer. We know that limk→∞ (R(k, k))/(k s ) = ∞ because R(k, k) ≥ (1+ o(1))(√2k/e)2k/2 . If R(k, k +1)/R(k, k) > 1+ c for some positive constant c, then when k tends to infinity, R(k, k + 1) − R(k, k) R(k, k + 1) − R(k, k) R(k, k) cR(k, k) = > →∞. ks R(k, k) ks ks Therefore, if the answer to Problem 3.1.2 is yes, then Conjecture 3.1.1 holds. As we cited earlier, Erdős thought that he could prove Conjecture 3.1.1 for s = 1, but no known proof can be found.

3.2 Some known results on diagonal Ramsey numbers In [96], Erdős proved R(k, k) > 2k/2 ; R(k, k) < 4k was proved in [110], a paper by Erdős and Szekeres. Erdős mentioned that this upper bound was proved by Szekeres (see [96]). The following Conjecture 3.2.1 (and the associated prize) was frequently mentioned by Erdős in his talks and problem papers. More complete descriptions of these and many other related problems in this vein can be found in the monograph Erdős on Graphs: His Legacy of Unsolved Problems by F. Chung, R. Graham [64].

20 | 3 Bi-color diagonal classical Ramsey numbers Conjecture 3.2.1. The limit limk→∞ R(k, k)1/k exists. The best known lower and upper bounds for R(k, k) currently stand at (1 + o(1))

c log k √2k k/2 2 ≤ R(k, k) ≤ k − log log k 4k , e

due to Spencer [273] and Conlon [70], respectively. Therefore, if the limit in Conjecture 3.2.1 exists, it must be between √2 and 4. Maybe some people like the limit in Conjecture 3.2.1 to be 2, but there seems little evidence now. Maybe it exists and equals √2, and the known upper bound on R(k, k) is very weak. It is natural to propose the following problem. Problem 3.2.1. Improve the known lower and upper bounds for R(k, k). Let us consider a small case, R(10, 10). We know that 798 ≤ R(10, 10) ≤ 23556. It is possible that the exact value is much smaller than the known upper bound 23,556. The lower bound was obtained by Paley graph G797 more than 30 years ago. Problem 3.2.2. Study the lower and upper bounds for R(10, 10).

3.3 Constructive lower bound for R(k, k) The best known lower bound (1 + o(1))(√ 2k/e)2k/2 ≤ R(k, k) in [273] was obtained by a probabilistic method based on the local lemma. A constructive lower bound for R(k, k) was given in [21], in which a (k, k)-graph of order 2n is constructed, where (log n)1−α0

k = 22

and α 0 is a positive constant smaller than 1. That is to say, we can obtain a constructive lower bound for α (log k)(log log k)

R(k, k) ≥ 2

0 1−α0

+1,

where α 0 is a positive constant smaller than 1. Gil Cohen obtained a better constructive lower bound for R(k, k) in his preprint [67] in June 2015. Before this work, the graphs showing that R(k, k) is greater than any polynomial of k were constructed via set-intersection, among which the first was constructed by Frankl. These constructive lower bounds are much smaller than the best known non-constructive lower bound for R(k, k) cited above. It is difficult to answer the following problem. Problem 3.3.1. Give R(k, k) a constructive lower bound no smaller than (1 + c)k for some constant c > 0.

3.4 R(5, 5) and almost regular Ramsey graphs

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Let c be a given positive constant. If we can construct a (k + 1, k + 1)-graph of order ⌈(1 + c)(R(k, k) − 1)⌉ based on a known (k, k)-Ramsey construction for any integer k ≥ 3, then we can give R(k, k) a constructive lower bound 5(1 + c)k−3 + 1. It is far from being reached. Some results of Alon in [5] explained the difficulties in finding an explicit construction for good Ramsey graphs defined by real polynomials. However, it is a problem if it can partially explain the difficulties in finding an explicit construction for good Ramsey graphs. We cannot confirm this before we consider other possible ways of finding an explicit construction. For instance, maybe Paley graphs are much better than those graphs defined by real polynomials. Of course, it is difficult to know the clique numbers of large Paley graphs. Even to give good lower and upper bounds seems difficult too. We will discuss Paley graphs further in Chapter 4.

3.4 R(5, 5) and almost regular Ramsey graphs A graph G is almost regular if ∆(G) − δ(G) ≤ 1. Let us consider R(5, 5) and the existence of almost regular Ramsey graphs in this section. The problems on almost regular Ramsey graphs in this section were proposed by Xiaodong Xu and were studied by Zehui Shao, Xiaodong Xu, and Linqiang Pan. Most of these problems can be found in [252]. Among small classical Ramsey numbers, R(5, 5) is one of the most interesting ones. It is the smallest diagonal classical Ramsey number of which the exact value is unknown. We know that 43 ≤ R(5, 5) ≤ 49 , which were given in [115] and [196], respectively. In [196], the following problems were discussed, and it was conjectured by Mckay and Radziszowski that the answer to Problem 3.4.1 is yes and that to Problem 3.4.2 is no. Problem 3.4.1. R(5, 5) = 43? The following two problems are related to Problem 3.4.1. We know 656 (5, 5)-graphs of order 42, and none of them is almost regular (see [196]). Therefore, if the answer to the following Problem 3.4.3 is yes, then it is yes for Problem 3.4.2 too. Problem 3.4.2. Is there a new (5, 5)-graph of order 42 other than the known 656 ones? Note that Mckay and Radziszowski [196] conjecture that the answer to Problem 3.4.1 should be yes and that the answer to Problem 3.4.2 should be no, and Exoo conjectures that the answers to both Problems 3.4.1 and 3.4.2 should be yes. We would like propose the following problem now. Problem 3.4.3. Is there an almost regular (s, t)-Ramsey graph for any positive integers s and t no smaller than 3?

22 | 3 Bi-color diagonal classical Ramsey numbers In Problem 3.4.3, if s = t, we still cannot solve the problem. We can also propose a more general problem similar to Problem 3.4.3, as in the following Problem 3.4.4. In particular, we do not know if there is an almost regular (5, 5)-Ramsey graph if we suppose that R(5, 5) = 43. Some related computations were done for small cases in [252]. Problem 3.4.4. Is there an almost regular (s, t)-graph of order n for any positive integers s, t, and n ≤ R(s, t)? Xiaodong Xu conjectures that the answer to this problem should be yes, but Radziszowski does not support such a conjecture unless it is restated only for sufficiently large s and t. Of course, even for sufficiently large s and t, we cannot prove it now. Alon, Ben-Shimon, and Krivelevich studied regular (3, t)-graphs of large orders in [8]. They proved that there is a positive absolute constant C such that for every positive integer n there exists a triangle-free regular graph with no independent set of size at least C(n log n)1/2 . Now, let us consider the relation between this result and almost regular Ramsey graphs. Let s ≥ 3. If there is an almost regular (s, t)-graph of order n, then there is a regular (s, 2t−1)-graph of order 2n. In fact, supposing that G0 is an almost regular (s, t)-graph of order n, and ∆(G) = δ(G) + 1 we can construct a regular (s, 2t − 1)-graph G of order 2n as follows. Suppose that both G1 and G2 are isomorphic to G0 , and V(G1 ) ∩ V(G2 ) = 0. Suppose that the set of vertices of degree δ(G0 ) in V(G1 ) and V(G2 ) are {u 1 , . . . , u t } and {v1 , . . . , v t }, respectively. Let V(G) = V(G1 ) ∪ V(G2 ) and E(G) = E(G1 ) ∪ E(G2 ) ∪ {u i v i | i ∈ [t]}. It is not difficult to see that G is a ∆(G0 )-regular (s, 2t − 1)-graph. Therefore, if the answer for Problem 3.4.4 is yes, then we can obtain a result similar to that in [8] cited above, which states that if the answer to Problem 3.4.4 is yes, then there is an (s, 2t − 1)-regular graph of order 2R(s, t) − 2. The following conjecture may be true with a high probability. Conjecture 3.4.1. For any ϵ > 0, there is a positive integer n = n(ϵ) such that: for any integers s and t no smaller than 3, if R(s, t) > n, then there is an (s, t)-Ramsey graph G such that ∆(G) − δ(G) 5, R(4, 4) > 17, R(6, 6) > 101, R(8, 8) > 281 and R(10, 10) > 797) or graphs constructed based on Paley graphs by Shearer’s method in Theorem 4.1.1 (for instance, R(7, 7) ≥ 205, R(9, 9) ≥ 565, R(11, 11) ≥ 1597, see [257]). Two small exceptions are the best known lower bounds for R(5, 5) and R(12, 12). R(5, 5) ≥ 43 was obtained by Exoo in [115], and R(12, 12) ≥ 1639 were obtained in [309] by R(11, 11) ≥ 1597 and the equality R(s, t + 1) ≥ R(s, t) + 2s − 2 for s ≥ 5. Recently, Milos Tatarevic proved that R(12, 12) ≥ 1640; the discussion can be found at https://mtatar.wordpress.com/ 2017/02/14/r12/. There may be a cyclic (12, 12)-graph of order no smaller than 1640, but it may be not easy to find such a graph. The best known lower bound on R(k, k) was obtained by Theorem 4.1.1 for any k between 13 and 22. We will discuss the multicolor version of Shearer’s construction in Section 6.5.

4.2 Some known results on the clique numbers of random graphs In this section, we will cite some known results on the clique numbers of random graphs [42]. Note that there is a section on Paley graphs in a book titled Random https://doi.org/10.1515/9783110576702-004

26 | 4 Paley graphs and lower bounds for R(k, k)

Graphs [42] written by Bollobás. As pointed out in [42], Paley graphs are related to random graphs in this way: Paley graph G p rather closely resembles a typical graph in G(p, 1/2), which consists of all graphs with vertex set {1, . . . , p} in which the edges are chosen independently and with probability 1/2. Suppose that p ∈ [0, 1]. This is in accordance with customary usage. Based on context, this probability p will not be confused with the prime p in Paley G p . Let us write b for 1/p. Let r0 = r0 (n, p) be the positive real number satisfying −r 0 −1/2 r 0 (r 0 −1)/2

f(r0 ) = (2π)−1/2 n n+1/2 (n − r0 )−n+r0 −1/2 r0

p

=1.

Therefore, r0 = 2 log b n − 2 log b log b n + 2 log b (e/2) + 1 + o(1). Then we have the following theorem proved in [41]. Theorem 4.2.1. For a.e. G ∈ Ç(N, p) there is a constant m 0 (G) such that if n ≥ m0 (G), then ⌊r0 (n) − 2(log log n)/ log n⌋ ≤ cl(G n ) ≤ ⌊r0 (n) + 2(log log n)/ log n⌋ . Furthermore, | cl(G n ) − 2 log b n + 2 log b log b n − 2 log b (e/2) − 1| < 3/2 . For instance, if n = 9973 and p = 0.5, then ⌈2 log b n − 2 log b log b n + 2 log b (e/2) + 1⌉ = 21 . On the other hand, cl(G9973 ) = 19 can be found in [216]. By the known data on cl(G p ) for p < 10,000, we can conjecture that G p does not resemble a typical graph in G(p, 1/2), because cl(G p ) seems to wave more seriously when prime p tends to ∞. In the following problem, we consider what happens when the prime p is large. Let the least quadratic non-residue of odd prime p be n(p). We know that cl(G p ) ≥ n(p), because {0, . . . , n(p) − 1} is a clique of order n(p). Burgess [127] proved that n(p) = O ϵ (p1/(4√e)+ϵ ) for any given positive constant ϵ. The generalized Riemann hypothesis (GRH) asserts that for every Dirichlet character χ and every complex number s with L(χ, s) = 0: if the real part of s is between 0 and 1, then it is actually 1/2. The generalized Riemann hypothesis (for Dirichlet L-functions) was probably formulated for the first time by Adolf Piltz in 1884. Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime numbers. It was proved in [16] that if GRH holds, then n(p) = O(log2 p). We can see that if some mathematician can prove GRH, then we may understand Paley graphs and even R(k, k) better. The best known results on the upper bounds for n(p) seem weak. We can tell very little about the upper bound for cl(G p ) better than ⌊p1/2 ⌋. In [192] it was proved that cl(G p ) ≤ (p − 4)1/2 . We know that (p − 4)1/2 is only a little smaller than p1/2 , and

4.3 Some problems and conjectures on Paley graphs |

27

(p − 4)1/2 > p1/2 − 1 for any odd prime p > 5. Some computation on the cases where p is not very large shows that this upper bound for cl(G p ) is often much larger than the exact value. For the lower bound on cl(G p ), a result in [146] shows that infinitely many Paley graphs of order p contain a clique of order c log p log log log p. Earlier results in [200] show that we have a lower bound c log p log log p infinitely often if GRH holds. Both of these results were obtained by studying n(p). See [62] for a related discussion. ¯ that n(p) = O(log p log log p). It is conjectured by Enrique Trevino Therefore, we may say that all interesting results on the lower bound for cl(G p ) were obtained by studying n(p). These known results are not enough for us to understand cl(G p ) deeply, because the gap between cl(G p ) and n(p) may be very large. On the other hand, as we know, the general upper bound for cl(G p ) is about p1/2 , which seems very weak. Let n be an integer and n ≥ 10 and Gc (n, q) denote the set of random cyclic graphs of order n in the following sense, where q ∈ (0, 1). For any integer i in {1, . . . , ⌊n/2⌋}, i ∈ S holds randomly with probability q. Let a cyclic graph of order n with parameter set S obtained this way be a random cyclic graph. It is interesting to know if the expectation of cl(G1 ) equals that of cl(G2 ) for any G1 ∈ Gc (n, q) and any G2 ∈ G(n, q). This problem does not seem to be easy.

4.3 Some problems and conjectures on Paley graphs As Shearer remarked in [257], the behavior of cl(G p ) for large p is unknown and appears to be a difficult problem. Shearer computed large p such that cl(G p ) = k for given k and p ≤ 3000. These data were extended to primes between 3000 and 10,000 based on the computational results given by Shearer himself (under 7000) [258] and Exoo (between 7000 and 10,000) [216]. We may do more computations to extend the data further. Shearer found that there seems to be a definite tendency for cl(G p ) to be odd, which he was unable to explain. If we construct a large graph of order 2p rather than 2p + 2 without vertices λ and 󸀠 λ in the construction of Shearer, must the clique number of the new large graph equal cl(G p ) + 1 or it may equal cl(G p ) in some cases? It was pointed out in [192] that the equality cl(G p ) = n(p) seems unlikely to happen very often. Indeed, for p ≤ 7000, it happens only six times (see [258]). In [192], Maistrelli and Penman proposed the following conjecture based on this fact. Conjecture 4.3.1. The equality cl(G p ) = n(p) occurs for only finitely many p. This is an interesting conjecture. However, by what we know now, it seems too early to conjecture whether cl(G p ) = n(p) occurs for only finitely many p. If p is a prime such that p ≡ 1 (mod 4), and p tends to ∞, then we write p (p) → ∞. We propose the following conjecture.

28 | 4 Paley graphs and lower bounds for R(k, k) Conjecture 4.3.2. The limit limp (p)→∞ (cl(G p ))/(p1/2 ) = 0. This is to say that we conjecture that for any positive ϵ > 0, there are only finite primes p such that cl(G p ) > ϵp1/2 . This is not a bold conjecture, but it is far from being reached now. Problem 4.3.1. Does limp (p)→∞ (cl(G p ))/(n(p)) = ∞ hold? Does limp (p)→∞ (cl(G p ))/ (p ϵ ) = 0 hold for any ϵ > 0? It is obvious that if limp (p)→∞ (cl(G p ))/(n(p)) = ∞ holds, then Conjecture 4.3.1 holds. Now we cannot even prove limp (p)→∞ inf (cl(G p ))/(p ϵ ) = 0. We can do very little to show that this holds with a high probability. We know that Vinogradov conjectured that limp (p)→∞ (n(p))/(p ϵ ) = 0. We can see that limp (p)→∞ (cl(G p ))/(p ϵ ) = 0 implies limp (p)→∞ (n(p))/(p ϵ ) = 0 conjectured by Vinogradov. A little weaker, the following conjecture may be easier than proving that limp (p)→∞(cl(G p ))/(n(p)) = ∞, if both of them hold. Note that if the following conjecture holds, then Conjecture 4.3.1 holds too. It is interesting to study the gap between cl(G p ) and n(p). As we know, in most cases the gap is large. We conjecture that cl(G p ) − n(p) tends to infinity when prime p tends to infinity. In other words, what we are conjecturing is as follows. Conjecture 4.3.3. For any positive integer m, there are finite primes p such that p ≡ 1 (mod 4) and cl(G p ) − n(p) < m. Let us propose the following problem. Problem 4.3.2. Is there is a positive integer N such that for any primes p and q, if p ≡ 1 (mod 4) and q ≡ 1 (mod 4), and p > q > N, then cl(G p ) > n(q) always holds? We will do some computations on the lower bound for cl(G p ) in the next section.

4.4 The lower bound for cl(G p ) based on the greedy algorithm If we compute a lower bound for cl(G p ) by the greedy algorithm, then we find a clique that contains A = {0, 1, . . . , n(p) − 1}, where |A| = n(p). If the vertices in A have no common neighbors, then the clique we find is A, and if they have common neighbors, then we obtain a clique of order larger than n(p) and containing A as a proper subset. Therefore, if cl(G p ) = n(p), then the vertices in A do not have common neighbors. In this section, we will do some computations to obtain more data related to Conjecture 4.3.1. Let R(p) in {1, . . . , p − 1} be the set of all quadratic residues modulo p. Let f(p) be the length of the longest arithmetic progression with common difference 1 in R(p). It is not difficult to prove the following simple results, which are interesting for us because

4.4 The lower bound for cl(G p ) based on the greedy algorithm

| 29

they relate Ramsey numbers to van der Waerden numbers indirectly through Paley graphs and quadratic residues of p. Theorem 4.4.1. If n(p) ≤ f(p) ≤ 2n(p) − 1, then cl(G p ) ≥ f(p) + 1; If f(p) ≥ 2n(p), then cl(G p ) ≥ 2n(p). Proof. Let m = f(p) and a, a + 1, . . . , a + m − 1 be an arithmetic progression with common difference 1 in R(p). So a > n(p). If n(p) ≤ f(p) ≤ 2n(p) − 1, then it is not difficult to see that {0, . . . , n(p) − 1, a + n(p) − 1, . . . , a + m − 1} is a clique of order m + 1. Therefore cl(G p ) ≥ f(p) + 1. Now let us consider the case f(p) ≥ 2n(p). It is not difficult to see that {0, . . . , n(p) − 1, a + m − n(p), . . . , a + m − 1} is a clique of order 2n(p). The results in the following theorem can be found in many books on number theory. Theorem 4.4.2. 2 is a quadratic non-residue of primes of the form 8a ± 3. −3 is a quadratic non-residue of primes of the form 6a + 5. 5 is a quadratic non-residue of primes of the form 10a ± 3. Similarly, 7 is a quadratic non-residue of primes of the form 14a + 3, 5, 6. Based on these results, we may obtain the following result by a simple computation. Theorem 4.4.3. Let p be a prime and p ≡ 1 (mod 4). If n(p) > 6, then p ≡ 1, 49

(mod 120) .

We computed n(p) and f(p) for every prime p between 7000 and 10,000 first. We found that n(p) > f(p) in 8 cases, but in all of these cases n(p) < cl(G p ). For instance, the smallest prime p > 10,000 such that n(p) > f(p) is 10,369, and n(10,369) = 11. Note that we know only R(10, 10) ≤ 23,556, which is not enough to prove cl(G10,369 ) > 11. To know whether cl(G10,369 ) > 11, we have to do some computations. It is easy to know that cl(G10,369 ) ≥ 18 by computing. Now, let us list a few computing results on the lower bound on cl(G p ) for some small p’s here. It is not difficult to compute n(p) and f(p) even for p that is about 108 . We may find all the odd primes bounded by 108 such that n(p) > f(p) holds, for which it is possible that n(p) = cl(G p ). Let T be the set of primes p such that p ≥ 10,000 and elements in {0, 1, . . . , n(p)− 1} have no common neighbor in G p . By computing the lower bound on cl(G p ) for p ∈ T, we obtain the following results. 1. For any prime p between 10,000 and 20,000, cl(G p ) ≥ 19 and n(p) ≤ 17. 2. For any prime p between 20,000 and 30,000, cl(G p ) ≥ 20 and n(p) ≤ 17. 3. For any prime p between 30,000 and 53,880, cl(G p ) ≥ 21 and n(p) ≤ 19. By more computation, we found that for any prime p between 10,000 and 100,000, if p ≠ 87,481, then cl(G p ) > n(p) holds.

30 | 4 Paley graphs and lower bounds for R(k, k) Suppose that odd prime p > 10,000 and p ≡ 1 (mod 4). There is an integer i such that 1000 ≤ p/2i < 2000. Let the subgraph of G p induced by the common neighbors of {0, 1, . . . , i−1} be H p (i). Hence, cl(G p ) ≥ cl(H p (i))+i. If p is not very large, then |H p (i)| is not much larger than p/2i , and it is not difficult to compute cl(H p (i)) or to obtain a good lower bound for it. Most of the above results were obtained by computing the exact values or lower bounds on cl(H p (i)). We have obtained more lower bounds by computations. None of these computations require a lot of time. We list these lower bounds in the following. We computed the lower bound for cl(H31,249 (4)), the clique number of the subgraph in G31,249 induced by the common neighbors of {0, 1, 2, 3}, and obtained a lower bound 21. We write cl(G31,249 ) ≥ 4 + 21 = 25 for this result. We will write more lower bounds in such a form. We obtained that cl(G483,289 ) ≥ 11 + 22 = 33. Note that n(483,289) = 31. Other results obtained include cl(H87,481 (5)) = 24 and cl(H515,761 (9)) = 28, where n(87,481) = 29 and n(515,761) = 37. We obtained that cl(G p ) ≥ 23 for p ∈ {47161, 47569, 48049, 53089}, and cl(G52,369 ) ≥ 22. Let us consider a large p as an example. For prime p = 9,7426,981, both the least primite root and the minimum quadratic non-residue equal 53. It is interesting to find a clique of order 54. One does not know whether it is easy unless one tries to compute it. Let j be a positive integer such that j < n(p). Let graph H(p, j) be the subgraph of G p induced by the set of vertices that has at least j neighbors among {0, 1, . . . , n(p) − 1}. We can study the lower bound on cl(G p ) by computing the exact value or lower bound on cl(H(p, j)) for some appropriate j. Note that if j is too large, then the order of H(p, j) will be small, and H(p, j) will be of little use in studying the lower bound for cl(G p )). However, if j is too small, then the order of H(p, j) will be large, and H(p, j) may be difficult to deal with. Although we may conjecture that the limit in the following problem exists and equals 0, we cannot give a proof. Problem 4.4.1. Let f(k) be the largest prime p such that p ≡ 1 (mod 4) and cl(G p ) ≤ k − 1. Does limk→∞ f(k)/R(k, k) exist? If it exists, is it 0? It is interesting to find other graph classes that may show a good lower bound for R(k, k) better than that based on Paley graphs. Now let us cite the last paragraph in the section on Paley graphs in Bollobás’ book Random Graphs [42], where he wrote that n(p) > c log p log log p infinitely often under GRH does not dash our hopes of using Paley graphs to obtain a good general bound for Ramsey numbers, and nevertheless it seems likely that for sporadic values of p the graphs give even better lower bounds for Ramsey numbers than do our random graphs. Now let us remark on the discussion of Bollobás above. As we pointed out above, all interesting results on the lower bound for cl(G p ) were obtained by studying n(p), and the best known general upper bound for cl(G p ) is about p1/2 , which seems poor.

4.4 The lower bound for cl(G p ) based on the greedy algorithm

| 31

However, if we want to obtain better lower bounds for R(k, k) based on G p , it is not enough to study only n(p), because cl(G p ) may be much larger than n(p). Note that we do not even know if they equal each other for infinite primes p (see Conjecture 4.3.1). We have to study the upper bound for cl(G p ). That is to say, we have to improve the poor known upper bound that is about p1/2 , which seems not easy. It is not difficult to know that G p is not similar to random graphs in G(p, 1/2), and cl(G p ) is often far from the expectation of the clique number of graphs in G(p, 1/2) which varies much more smoothly. We do not know if it is NP-complete to compute the exact value of cl(G p ). In Paley graph G p , let U be a clique. If |U| is small, then the number of common neighbors of the vertices in U is near for different U of same order. However, if |U| is not small, then the number of common neighbors of the vertices in U may be not near for different U’s of the same order. Now we will consider a case that is interesting for us in studying cl(G p ), in which |U| is {0, . . . , n(p) − 1}. If {0, . . . , n(p) − 1} have no common neighbors in G p , what conclusion on the relation between n(p) and p can we obtain? It is not difficult to see that n(p) cannot be very small, otherwise {0, . . . , n(p) − 1} must have common neighbors in G p . Let us propose the following conjecture. Conjecture 4.4.1. There is a positive constant c, such that n(p) ≥ c log p when {0, . . . , n(p) − 1} have no common neighbors in G p . This conjecture is not a bold one, but we cannot prove it. We can find some primes with large least quadratic residues. We know that n(118,801) = 13, n(3,764,401) = 17, and n(8455561) = 19.

5 Bi-color off-diagonal classical Ramsey numbers In this chapter, we will discuss bi-color off-diagonal Ramsey numbers and the related Ramsey graphs. In some cases, we will discuss some results on off-diagonal multicolored classical Ramsey numbers in this chapter, if these results are similar to the related results on R(s, t). We will discuss R(3, s + 1) − R(3, s) in this chapter. Related problems and methods may be similar to those discussed in Chapter 3.

5.1 Some known bounds for off-diagonal Ramsey numbers In this section, we survey some problems and results on R(3, k) and other bi-color off-diagonal classical Ramsey numbers. The research of R(3, k) has a long history of more than half a century and includes the work of Erdős on the lower bound and the work of Ajtai, Komlós, and Szemerédi on the upper bound in [4]. Spencer wrote an article on the history of R(3, k) until 2009 (see [271]). We know 1 k2 k2 ( + o(1)) ≤ R(3, k) ≤ (1 + o(1)) . 4 log k log k The lower bound was obtained by Tom Bohman and Peter Keevash [40], and by Gonzalo Fiz Pontiveros, Simon Griffiths, and Robert Morris [120] independently and simultaneously. As pointed out in [40], both proofs exploit self-correction, but are different in some important ways. Note that Kim was the first mathematician to prove that R(3, k) ≥ ck2 /(log k) holds for some positive constant c. The upper bound is implicit in a 1983 paper by Shearer [256]. There are some references in which R(3, k) ≤ ck 2 /(log k) in [4] was cited, and Shearer’s work and [256] were not cited. This seems unfair in some cases. For the lower bound on R(4, k), in [37] Bohman proved that R(4, k) ≥ c

k5/2 log2 k

.

The following best known lower and upper bounds for R(s, t) were obtained in [39] and [180], respectively, c󸀠s

t

s+1 2

(log t)

s+1 1 2 − s−2

≤ R(s, t) ≤

(1 + o(1))t s−1 , (log t)s−2

where the lower bounds obtained in [39] hold for s ≥ 5.

https://doi.org/10.1515/9783110576702-005

34 | 5 Bi-color off-diagonal classical Ramsey numbers

5.2 Some problems and results on off-diagonal Ramsey numbers There are some constructive results on the lower bounds for off-diagonal Ramsey numbers. The following Theorem 5.2.1 and its multicolor generalization in Theorem 5.2.2 were proved in [312]. Theorem 5.2.1 ([312]). If k ≥ 2, s ≥ 5, then R(2k − 1, s) ≥ 4R(k, s − 1) − 3. Theorem 5.2.2 ([312]). For k 1 ≥ 5 and k i ≥ 2, i ≥ 2, we have R(k 1 , 2k 2 − 1, k 3 , . . . , k r ) ≥ 4R(k 1 − 1, k 2 , . . . , k r ) − 3 . A special case of Theorem 5.2.1 is R(5, s) ≥ 4R(3, s − 1) − 3 . By the best known upper bound for R(3, s − 1) we can see that 4R(3, s − 1) − 3 ≤ (4 + o(1))

(s − 1)2 . log(s − 1)

On the other hand, based on the result in [39] we know that R(5, s) ≥ c

s3 8

log 3 s

.

Therefore

R(5, s) =∞. 4R(3, s − 1) − 3 That is to say, R(5, s) ≥ 4R(3, s − 1) − 3 is weak for large s. For small s the best lower bound on R(5, s) may be obtained by 4A − 3, where A is the best known lower bound on R(3, s − 1). For instance, we know that R(3, 20) ≥ 111, based on which we have R(5, 21) ≥ 441, which is the best known lower bound on R(5, 21). On the other hand, for any integer s such that 5 ≤ s ≤ 20, the best known lower bound on R(5, s) is not obtained by R(5, s) ≥ 4R(3, s − 1) − 3, because for small s it is often easier to obtain a better lower bound for R(5, s) by computing. However, for k ≥ 4 and lim

s→∞

R(2k − 1, s) ≥ 4R(k, s − 1) − 3 , the problem is different, because the best known lower bound on R(2k − 1, s) and the best known upper bound on R(k, s − 1) seem weak. For instance, we cannot improve R(7, s) ≥ 4R(4, s − 1) − 3 similar to R(5, s) ≥ 4R(3, s − 1) − 3. In [63], F. R. K. Chung, R. Cleve, and P. Dagum proved that R(3, 4k + 1) ≥ 6R(3, k + 1) − 5 . Let us consider the following conjecture related to the known bounds for R(3, k).

5.2 Some problems and results on off-diagonal Ramsey numbers

| 35

Conjecture 5.2.1. The limit limk→∞ (R(3, k) log k)/(k 2 ) exists. By the known bounds for R(3, k), we know that if the limit in Conjecture 5.2.1 exists, it must be between 1/4 and 1. Some people believe that the best known upper bound for R(3, k) cited above is near its exact value. That is to say, they are inclined to conjecture that the limit in Conjecture 5.2.1 exists and equals 1. We are not sure whether we should think so. Maybe the limit exists, and equals neither 1/4 nor 1. If the limit exists but is smaller than 1, it may be much larger than 1/4 and near 1. Now let us discuss the gap between R(3, k + 1) and R(3, k). The following conjecture can be found in [64]. Conjecture 5.2.2. The limit limk→∞ (R(3, k + 1) − R(3, k)) = ∞. By the best known lower bound for R(3, k), it is not difficult to know that if the limit in Conjecture 5.2.2 exists, then it must be ∞. However, we cannot prove its existence. Note that even if we can prove that Conjecture 5.2.1 is true, we may still know very little about R(3, k+1)−R(3, k). This is similar to that even if we can prove the Prime number theorem, we may still not know much about the gap between consecutive primes. It is a natural question to ask whether we can obtain a large (3, k + 1)-graph by extending some (3, k)-Ramsey graph. Problem 5.2.1. For a (3, k)-Ramsey graph G, let f(G) be the maximum order of a (3, k + 1)-graph with an induced subgraph isomorphic to G. Let F(k) = maxG∈A k f(G), where A k is the set of all (3, k)-Ramsey graphs. Answer the following questions. lim (F(k) − R(3, k)) = ∞ ?

k→∞

lim (R(3, k + 1) − F(k)) = ∞ ?

k→∞

lim

k→∞

R(3, k + 1) − F(k) =0? k

We cannot prove if the limits in Problem 5.2.1 exist. We can prove that F(k)−R(3, k) ≥ 3, because we can construct a (3, k +1)-graph of order R(3, k)+2 which contains a (3, k)Ramsey graph as an induced subgraph. It is easy to see that R(3, k + 1) − F(k) ≤ R(3, k + 1) − R(3, k) ≤ k + 1 . More generally, we may consider an (s, k)-Ramsey graph and similar problem for integer s ≥ 3. Because the problem seems very difficult and similar to the special case s = 3 in Problem 5.2.1, we will not discuss more related details here. More generally than Conjecture 5.2.2, we may consider the following problem. Problem 5.2.2. For any given integer s ≥ 3, lim (R(s, k + 1) − R(s, k)) = ∞?

k→∞

36 | 5 Bi-color off-diagonal classical Ramsey numbers

It seems that even if we can prove that lim R(s0 , k + 1) − R(s0 , k) = ∞

k→∞

for some integer s0 ≥ 3, we still do not know how to prove lim R(s0 + 1, k + 1) − R(s0 + 1, k) = ∞

k→∞

based on it, unless the method used on s0 works on general s too. We can also propose the following problem. If the correct answer is yes, then the exact value of R(s, k) must be near the best known upper bound. Problem 5.2.3. For any given integer s ≥ 3 and any ϵ > 0, lim

k→∞

R(s, k + 1) − R(s, k) =∞? k s−2−ϵ

The following problem is one of the most important problems in Ramsey theory. Problem 5.2.4. Improve the known lower and upper bounds for R(s, t). Conjecture 5.2.3. There are absolute constants c1 and c2 such that 0 < c1 ≤

R(3, k + 1) − R(3, k) R(3,k) k

≤ c2 .

Furthermore, for given integer s ≥ 3, 0 < a1 (s) ≤ (

R(s, k + 1) − 1) k ≤ a2 (s) , R(s, k)

where a1 (s) and a1 (s) are constants relative only to s. It is not difficult to see that if 0 < a1 (s) ≤ (

R(s, k + 1) − 1) k ≤ a2 (s) R(s, k)

in Conjecture 5.2.3 holds, we still cannot solve Problem 5.2.3, i.e., whether lim

k→∞

R(s, k + 1) − R(s, k) =∞ k s−2−ϵ

holds, because the best known lower bound on R(s, k) is not enough to prove it. If the first part of Conjecture 5.2.3 holds, i.e., there are absolute constants c1 and c2 such that R(3, k + 1) − R(3, k) 0 < c1 ≤ ≤ c2 , R(3,k) k

then we may further ask if lim

n→∞

R(3, k + 1) − R(3, k) R(3,k) k

exists. Maybe this limit exists and equals 2. If this does not hold, then R(3, k + 1) − R(3, k) may wave seriously.

5.3 Lower bounds for some small off-diagonal Ramsey numbers

| 37

Problem 5.2.5. Does limn→∞ (R(3, k + 1) − R(3, k))/(R(3, k)/k) exist? There are many problems on prime numbers, some of which are solved and others are open. The prime counting function π(n) is defined as the number of primes not larger than n. The distribution of primes in the large range, such as the question how many primes are smaller than a given, large threshold, is described by the Prime number theorem, which states that π(x) lim x = ∞ . x→∞

ln x

There are some works on the difference of the consecutive prime numbers. We may consider setting similar problems on R(3, k) and ∆ k . However, the solutions to these problems may be very different from those proved or conjectured for the problems on prime numbers. If R(3, k + 1) − R(3, k) waves seriously, then it is similar to the famous weaker version of the twin prime conjecture proved by Yitang Zhang, which states that there is a constant c such that there are infinite pairs of consecutive primes such that p i+1 −p i < c. However, we do not believe that R(3, k + 1) − R(3, k) waves seriously in infinitely many cases. Note that we even do not know if there is only finite positive integer k such that R(3, k + 1) − R(3, k) = 3 holds. We believe that R(3, k + 1) − R(3, k) varies much more smoothly, different from the gap between consecutive primes. Some people consider the following conjecture, but no one can prove it. Among these people, Wang Rui gave a false “proof” [293], and the mistake cannot be corrected to obtain any new results. Conjecture 5.2.4. If a and b are integers and a ≥ b ≥ 4, then R(a, b) ≥ R(a + 1, b − 1) .

5.3 Lower bounds for some small off-diagonal Ramsey numbers Similar to R(5, 5), R(3, 10) is another interesting small classical Ramsey number. In [138], R(3, 10) ≤ 42 was proved. On the other hand, we know that R(3, 10) ≥ 40. Problem 5.3.1. R(3, 10) = 40? The following problem may be much easier, but it has not been solved either. Problem 5.3.2. Is there a 9-regular (3, 10)-graph of order 40? In the preprint titled “Computational lower limits on small Ramsey numbers”, Eugene Kuznetsov obtained new lower bounds for some bi-color classical Ramsey numbers. Based on new lower bounds R(5, 11) ≥ 183 and R(6, 9) ≥ 183 among them, we can obtain R(9, 12) ≥ 729 and R(7, 17) ≥ 729 by using R(2k − 1, l + 1) ≥ 4R(k, l) − 3 .

38 | 5 Bi-color off-diagonal classical Ramsey numbers

Tab. 5.1: Simple lower bounds for some small Ramsey number R(s, t). t s 6 7 8 9 10

10

331

11

457 609 816

12

14

865 965

327 549 831 881 983

471 729 847

13

15

585 1125

Tab. 5.2: Simple lower bounds for more small Ramsey number R(s, t). t s 4 5 6 7 8

16

17

18

19

20

21

22

23 319

888 617 875

729 965

920 1250

1226 1631

1080 1253 1665

Most new lower bounds in Tables 5.1 and 5.2 were obtained similarly based on this inequality or other inequalities. The best known lower bounds for some small Ramsey numbers were obtained by cyclic colorings. In [231], Radziszowksi, and Kreher proved that there is no cyclic (6, 6)-graph of order 100, 102, or 103. The lower bounds for the Ramsey number R(s, t) that can be obtained by cyclic colorings were studied by Harborth and Krause in [161] for any s, t such that 3 ≤ s ≤ 8 and 3 ≤ t ≤ 15, with the lower bound no larger than 102. In [77], the lower bounds for the Ramsey number R(3, t) that can be obtained by cyclic colorings were studied further with the lower bound no larger than 127. In [292], Wang Qingxian proved that R(3, k) ≥ 7k − 16 by constructing cyclic (3, k)-graphs, where k is an odd integer and k ≥ 7. This inequality is weak for large k. It is interesting to prove a better general lower bound for R(3, k) based on cyclic graphs. More importantly, it is interesting to know whether the order of the largest (3, k)-cyclic graph matches R(3, k) − 1 for large k. For R(s, k), we may consider a similar problem. Problem 5.3.3. Suppose that n is an integer and n ≥ 128. Study the best lower bound for R(3, k) given by K3 -free cyclic graphs of order n. To obtain a new cyclic lower bound for R(4, k) is much more difficult than for R(3, k), because in most cases a (4, k)-graph has larger minimum degree and a more complex

5.5 On R(3, s) and R(K3 , K s − e) |

39

structure. There may be more results on cyclic lower bounds on R(s, k) for s ≥ 4 in the future.

5.4 Upper bounds for bi-color off-diagonal classical Ramsey numbers We know that R(s, t) ≤ R(s − 1, t) + R(s, t − 1) . We also know that if both R(s − 1, t) and R(s, t − 1) are even, then the inequality is strict. We cannot improve this inequality in the general case now. Let us cite only one theorem on the upper bounds for bi-color classical Ramsey numbers in [166]. Theorem 5.4.1. Let m ≥ 4, n ≥ 4, R(m−2, n) ≤ α+1, R(m, n−2) ≤ β+1, and parameter x ∈ (0, 3). Also, let f(x, y) = A + √A2 − B , A=

3(y + α − β) − 2(1 + α)x , 9 − 4x

g(x, y) = A − √A2 − B , B=

(3 − x)(y + α − β)2 + xy2 . (3 − x)(9 − 4x)

Then, (a)

R(m, n) ≥ 2 + f(x, y)

or

R(m, n) ≤ 2 + g(x, y)

if 0 < x < 9/4 ;

(b)

R(m, n) ≤ 2 + f(x, y)

if x ∈ (9/4, 3) ;

(c)

R(m, n) ≥ α + β + 4 + 2/3√(α + 2β + 3)(2α + β + 3) + (β − α)2

if x = 9/4 .

The best known upper bounds for many small classical Ramsey numbers were obtained by Theorem 5.4.1 and R(s, t) ≤ R(s − 1, t) + R(s, t − 1) in [166], including R(5, 12) ≤ 848 and R(5, 14) ≤ 1461. These upper bounds may be much larger than the exact values, but they are the best known ones.

5.5 On R(3, s) and R(K 3 , K s − e) We will discuss R(3, s) and R(K3 , K s − e) in this section. Most results and problems and conjectures are cited from [318]. Let ∆ s = R(3, s) − R(3, s − 1). We will consider the upper and lower bounds for ∆ s . We know that 3 ≤ ∆ s ≤ s. By Theorem 3.1.1 in Chapter 3, we know that R(3, s + t + 1) ≥ R(3, s + 1) + R(3, t + 1) + s − 1 , where 2 ≤ s ≤ t. The lower bound for R(3, s + t) in the following Theorem 5.5.1 was proved by Gyárfás, Sebõ, and Trotignon based on this inequality in [155] in 2012, and was later proved in [318] based on a different constructive method.

40 | 5 Bi-color off-diagonal classical Ramsey numbers Theorem 5.5.1. If 3 ≤ s ≤ t, then R(3, s + t) ≥ R(3, s + 1) + R(3, t + 1) − 3. In [155] this theorem was proved in the following way. Since R(3, s + t) ≥ R(3, s) + R(3, t+1)+s−2 and R(3, s+1) ≤ R(3, s)+s+1, we have R(3, s+t) ≥ R(3, s+1)+R(3, t+ 1)−3. Further, if R(3, s+ t) = R(3, s+1)+ R(3, t+1)−3, then R(3, s+1) = R(3, s)+ s+1, and every (3, s + 1)-Ramsey graph must be s-regular. Now let us consider a similar result on R(3, s + t −1) proved constructively in [318], of which the proof is similar to that of Theorem 5.5.1 given in [318]. Theorem 5.5.2. For s, t ≥ 3, given any (3, s+1; m)-graph G which has two non-adjacent vertices with at most c G common neighbors, and any (3, t + 1; n)-graph H which has two nonadjacent vertices with at most c H common neighbors, we can construct a (3, s+t−1)graph of order m + n − c G − c H − 4. Therefore, we have R(3, s + t − 1) > m + n − c G − c H − 4 . Suppose n > 3. It is not difficult to see that in most cases, c G and c H above need not be large. In fact, if G is a d-regular (3, s + 1)-graph of order n, then for any vertex u ∈ V(G), there is a non-neighbor v such that u and v have at most ⌊(d(d − 1))/(n − d − 1)⌋ common neighbors. If we suppose that G is a (3, s + 1)-graph, we can obtain ∆ s + ∆ s−1 ≤ 2s −

(s − 1)(s − 2) R(3, s) − s − 1

similarly. Note that ∆ s ≤ s. Now let us consider the lower bound on R(3, s + 1). Theorem 5.5.3. Suppose that G is a (3, s)-graph. If α(G − e) = s − 1 for some edge e ∈ E(G), then R(3, s + 1) > |V(G)| + 4 . Note that the proof of Theorem 5.5.3 is similar to that of Theorem 5.5.2, and a known (3, 4)-Ramsey graph of size 10 is used in the construction. Later we found that Theorem 5.5.3 can be proved directly. It is interesting to know whether R(3, s) > R(K3 , K s − e) for every s ≥ 3. It holds if 3 ≤ s ≤ 11. We know that 52 ≤ R(3, 12) ≤ 59 and 47 ≤ R(3, 11) ≤ 50, and there is no interesting known lower bound on R(K3 , K12 − e) larger than 47. In [318], the following theorem was proved. Theorem 5.5.4. If s ≥ 2, then R(3, s + 1) ≥ R(K3 , K s − e) + 4 . We know that R(3, s + t − 1) ≥ R(3, s) + R(3, t) − 1. In [318] the following theorem was proved.

5.6 Why are these problems on R(3, k) so difficult? | 41

Theorem 5.5.5. If s, t ≥ 3, then R(3, s + t − 1) ≥ R(K3 , K s+1 − e) + R(K3 , K t+1 − e) − 5 . Maybe there exists a positive integer N0 such that ∆ i − ∆ i+1 ≤ 0 for any i ≥ N0 . Instead of being too bold, Rujie Zhu, Xiaodong Xu, and Radziszowski proposed the following conjecture in [318]. Conjecture 5.5.1. There exists a constant d ≥ 2 such that for all s ≥ 2, we have ∆ s − ∆ s+1 ≤ d . It seems that even if we can prove Conjecture 5.5.1, we still do not know how to prove limk→∞ ∆ k = ∞. On the other hand, the following theorem was proved in [318]. Theorem 5.5.6. If Conjecture 5.5.1 holds, then limk→∞ ∆ k /k = 0. If we want to be less bold, we may conjecture more softly. It is not difficult to prove the following theorem similarly by a more detailed analysis. Theorem 5.5.7. Suppose that a, b ∈ (0, 1) and a+2b < 1. If there exist positive integers c and N0 such that for any i ≥ N0 , ∆ i − ∆ i+1 ≤ c(log i)a , then lim

k→∞

∆k k (log k)b

=0.

We cannot prove the following conjecture, which may be easier than proving limk→∞ ∆ k = ∞. Conjecture 5.5.2. There exists a positive integer k such that lim Σ ki=0 ∆ s+i = ∞ .

s→∞

5.6 Why are these problems on R(3, k) so difficult? In this section, we will discuss the question in the section heading. As mentioned in last section, even if we can prove Conjecture 5.5.1, we still do not know how to prove limk→∞ ∆ k = ∞. We can prove the following theorem. Theorem 5.6.1. If both Conjecture 5.5.1 and Conjecture 5.5.2 hold, then lim ∆ s = ∞ .

s→∞

Proof. For any M0 > (k(k + 1)d)/2, let M = M0 /(k + 1) − (kd)/2. If Conjecture 5.5.2 holds, then there is positive integer s0 such that for any s ≥ s0 , M0 ≤ ∆ s + ⋅ ⋅ ⋅ + ∆ s+k . If Conjecture 5.5.1 holds, then ∆ s + ⋅ ⋅ ⋅ + ∆ s+k ≤ (k + 1)∆ s+k + (k(k + 1)d)/2. Therefore, ∆ s+k ≥ 1/(k + 1)((M0 − k(k + 1)d)/2) = M0 /(k + 1) − (kd)/2 = M, and we have lims→∞ ∆ s = ∞.

42 | 5 Bi-color off-diagonal classical Ramsey numbers Furthermore, if Conjecture 5.5.1 holds, then it is impossible that s − ∆ s is very small, because ∆ s +∆ s−1 is much smaller than 2s. Therefore, Conjecture 5.5.1 is not such weak a conjecture as it looks. In fact, we can consider the details as follows. If ∆ s = s, then every (3, s)-Ramsey graph G is an s − 1-regular graph. For any vertex v ∈ V(G), it is not difficult to prove that there is a non-neighbor of v, say u, such that v and u have at least a = ⌈((s − 1)(s − 2))/(R(3, s) − 1 − s)⌉ common neighbors. Therefore, ∆ s + ∆ s−1 ≤ 2s − a − 1, and we know that ∆ s−1 ≤ s − a − 1 based on the construction of G. Hence, ∆ s − ∆ s−1 ≥ a + 1. However, we know that a = ⌈((s − 1)(s − 2))/(R(3, s) − 1 − s)⌉ ≥ (1 − o(1)) log s based on the known bound for R(3, s). Hence, we have the following theorem. Theorem 5.6.2. If Conjecture 5.5.1 holds, then there is a positive integer s0 such that ∆ s < s for any integer s ≥ s0 . Some results on the upper bound for small R(3, k) were obtained by considering the edge numbers in (3, k)-graphs of order n. We have to say, when no one can prove general theorems on these difficult problems, people should be encouraged to obtain some useful data by computing. Some algorithms designed this way may be of more value than we expect. In some cases, proving small results and improving them may lead us towards important results. Maybe we can learn more from physicists. If some experimental physicists do good jobs, theoretical physicists will acknowledge their contribution. Of course, what Arnold and Serre believe is different for these problems. Furthermore, any useful idea related to R(3, k) may be helpful in improving related inequalities on multicolored classical Ramsey numbers too, about which we know even less. For instance, what we know about the upper bound for R m (3) is nearly the same as Schur knew in 1916. The best known upper bound on R5 (3) is 307, which can be obtained by R m (3) ≤ m(R m−1 (3) − 1) + 2 and R4 (3) ≤ 62. The exact value of R5 (3) may be near the best known lower bound 162. Can we improve the lower bound on R5 (3) based on a (3, k)-graph G of order n? Here, n ≥ 162 and k = 27. Note that we know that R(3, 27) ≥ 167. Let the known (3, 27)-graph of order 166 be G. Then, for G and any induced subgraph with order 162, the maximum independence number is no larger than 26, which is smaller than ⌊(162 − 1)/5⌋ = 32. Before working on lim k→∞ ∆ k /k = 0, we may consider whether there is an integer k 0 such that k > ∆ k holds for any k > k 0 . We will discuss why it is difficult later. It is a more difficult problem to prove that lim (k − ∆ k ) = ∞ .

k→∞

All of which are out of reach now.

5.6 Why are these problems on R(3, k) so difficult? | 43

5.6.1 Another idea Now, let us consider the following idea, which is not enough to improve ∆ k ≥ 3. Let G be a (3, s)-graph of order n, and V(G) = {u 1 , . . . , u n }. Suppose that u 2 and u 3 are two neighbors of u 1 in G. Let U i be the set of neighbors of u i in G for any i ∈ {1, 2, 3}. Let us construct graph H as follows. Let V(H) = V(G) ⋃{v, v1 , v2 , v3 }. Let E0 = {(v, v i ) | 1 ≤ i ≤ 3} and E i = {(u, v i ) | u ∈ U i } for any i ∈ {1, 2, 3}. Let E(H) = E(G) ⋃ ⋃3i=0 E i . It is not difficult to prove that if u 2 and u 3 are not in any s − 1-independent set in G, then α(H) ≤ s and R(3, s + 1) ≥ |V(G)| + 4. Hence, if u 2 and u 3 are not in any s − 1independent set in G, and G is a (3, s)-Ramsey graph, then R(3, s + 1) ≥ R(3, s) + 4. In fact, if α(H) ≥ s and A is an s-independent set in H, then it is not difficult to see that |A ⋂{v1 , v2 , v3 }| is 2 or 3, and v is not in A. We can prove the following theorem case by case. Theorem 5.6.3. Suppose that G is a (3, s)-Ramsey graph. If u 2 and u 3 are two neighbors of u 1 in G, and u 2 and u 3 are not in any s − 1-independent set in G, then R(3, s + 1) ≥ R(3, s) + 4 . We can see that if there is such a graph G, then the degree of u 1 in G must be smaller than s − 1, otherwise any two neighbors of u 1 are in an s − 1-independent set.

5.6.2 What makes ∆ s < s difficult to prove? We believe that it is important to study whether limk→∞ ∆ k /k = 0 holds. It is difficult to prove an interesting upper bound for ∆ s . In fact, we know only ∆ s ≤ s, which is trivial. We cannot prove that there is a positive integer s0 such that ∆ s < s for any s ≥ s0 , which may be considered as the simplest non-trivial case in improving the basic general inequality R(s, t) ≤ R(s − 1, t) + R(s, t − 1) . As we know, how to improve this basic inequality is one of the most important and basic problems in Ramsey theory. Any improvement may be important for us in understanding classical Ramsey numbers better. This makes us remember what Tutte said on the Four-color theorem. He proclaimed that the four-color problem is the tip of the iceberg, the thin end of the wedge, and the first cuckoo of spring.

5.6.3 What makes ∆ s > 3 difficult to prove? On the other hand, it seems difficult to prove an interesting lower bound for ∆ s constructively. How to study the lower bound for ∆ s by probabilistic methods? It would seem that we remain fully ignorant about this problem, at least the authors remain so.

44 | 5 Bi-color off-diagonal classical Ramsey numbers We know that ∆ s ≥ 3 was proved for any s ≥ 2 in 1989, but even today we do not know whether there is a positive integer s0 such that ∆ s ≥ 4 for any s ≥ s0 . Although we prove Theorem 5.6.3 in this chapter, it is not enough to prove ∆ s ≥ 4. In the following, we prove no theorem and only discuss why it is difficult to prove ∆ s ≥ a for positive integer a that is not very small. Let us discuss the minimum degree of a (3, s)-Ramsey graph G first. We know that ∆ s ≤ δ(G) + 1. We can see that δ(G) ≥ 2 because ∆ s ≥ 3. However, we do not know whether there is a positive integer s0 such that δ(G) > 2 for any s ≥ s0 and any (3, s)Ramsey graph G. If there is such an s0 , then we have R(3, s) ≥ R(3, s −1)+4 for s ≥ s0 , because we can prove R(3, s) ≥ R(3, s − 1)+ 3 by constructing a (3, s)-graph G of order R(3, s − 1) + 2 with δ(G) = 2 based on the method in [313]. Of course, for every edge maximal (3, s)-Ramsey graph G, δ(G) cannot be too small, because every non-adjacent pair of vertices share at least one common neighbor, and δ(G)(δ(G) − 1) + δ(G) + 1 ≥ R(3, s) − 1 . Therefore, δ(G) ≥ √ R(3, s) − 2. Hence, by the best known lower bound on R(3, s) we have 1 s δ(G) ≥ ( − o(1)) 2 √log s for every edge maximal (3, s)-Ramsey graph G, and δ(G) tends to ∞ if s tends to ∞. In [309], it was proved that R(k, s + 1) ≥ R(k, s) + 2k − 2 for k ≥ 5, by constructing a (k, s + 1)-graph G of order R(k, s) + 2k − 3, in which V(G) = V1 ⋃ V2 , V1 ⋂ V2 = 0 such that G[V1 ] is a (k, s)-Ramsey graph and G[V2 ] contains no independent set of order 3, where |V2 | = 2k − 2. Hence, to obtain better construction, it is interesting to consider the following problem, which is an important difficulty we now have. Problem 5.6.1. Given a general constructive method to construct a (k, s + 1)-graph G, in which V(G) = V1 ⋃ V2 , V1 ⋂ V2 = 0 such that G[V1 ] is a (k, s)-Ramsey graph and G[V2 ] is an independent set of order 3. In the construction above Theorem 5.6.3, we considered this problem for k = 3, but cannot prove that R(3, s + 1) ≥ R(3, s) + 4 this way. We can generalize Theorem 5.6.3 to the case k ≥ 3. That is to say, we can construct a (k, s + 1)-graph G mentioned in last paragraph in the case similar to that in Theorem 5.6.3, and how to do this in the general case, is an important difficulty that we mentioned above. Of course, we may also try to prove ∆ s > 3, by constructing a (3, s)-graph G, in which v is a vertex in V(G), and the subgraph in G induced by the non-neighbors is a (3, s − 1)-graph of which the order is smaller than R(3, s − 1) − 1. To construct a (3, s)graph of order R(3, s − 1) + 3, we need d(v) to be larger than 3. This seems difficult in the general case too.

5.7 When does R(l, s + t − 2) ≥ R(l, s) + R(l, t) − 1 holds |

45

5.7 When does R(l, s + t − 2) ≥ R(l, s) + R(l, t) − 1 holds Let us discuss a good idea related to a bad “proof”. Liu Fugui gave an incorrect proof of R(l, s+t−2) ≥ R(l, s)+R(l, t)−1 for l, s, t all no smaller than 3. Zhang Zhongfu pointed out an obvious counter-example for l = s = t = 3, in which R(3, 3+3−2) = R(3, 4) = 9 and R(3, 3) + R(3, 3) − 1 = 6 + 6 − 1 = 11. Zhang suggested considering the inequality for l, s, t ≥ 3 such that s + t ≥ 7. Bo Yuehua pointed out that if l = 4, s = 3, t = 4, then we obtain a counter-example. Now we may ask, what about s + t ≥ 8? We may consider the case in which l = 3, s = 4, t = 4 or l = 3, s = 5, t = 3 first. We can see that R(3, 6) ≥ R(3, 4) + R(3, 4) − 1, i.e., R(l, s + t − 2) ≥ R(l, s) + R(l, t) − 1 holds when l = 3, s = 4, t = 4. However, R(3, 6) = 18 < 19 = R(3, 5) + R(3, 3) − 1, i.e., R(l, s + t − 2) ≥ R(l, s) + R(l, t) − 1 does not hold when l = 3, s = 5, t = 3. Note that if R(l, s + t − 2) ≥ R(l, s) + R(l, t) − 1, where l = 3, t = 3 and s ≥ 4, then R(3, s + 1) ≥ R(3, s) + 5. Now we cannot prove that R(3, s + 1) ≥ R(3, s) + 4, and have only R(3, s + 1) ≥ R(3, s) + 3. Similarly, if R(l, s + t − 2) ≥ R(l, s) + R(l, t) − 1 holds when l = 3 and t = 4, then we have R(3, s + 4 − 2) = R(3, s + 2) ≥ R(3, s) + R(3, 4) − 1 = R(3, s) + 8. We can now only prove that R(3, s + 2) ≥ R(3, s) + 7. As we can see, R(l, s + t − 2) = R(l, s) + R(l, t) − 1 holds for l = 3 and s = 4, t = 3. However, even so, we have to consider the larger s + t for the following reason. Suppose that l = 3, s = 4, t = 3. Therefore, R(l, s + t − 2) = R(3, 5) = 14. We know there is a unique (3, 5)-Ramsey graph of order 13, G13 , the generalized Paley graph defined by the cubic residues modulo 13. For any two disjoint vertex subset of V(G13 ), V1 and V2 , such that |V1 | = 5 and |V2 | = 8, it is not difficult to know that it is impossible that the subgraphs of G13 induced by V1 and V2 are a (3, 3)-graph and a (3, 4)-graph, respectively, because we know that a result on the vertex Folkman number, which will be discussed later in this book, states that in any red-blue vertex coloring of the complement graph of G13 , there must be either a red K3 or a blue K4 . Therefore, in this case, we cannot construct an (l, s+t−2)-graph based on an (l, s)graph and an (l, t)-graph by adding edges. Let us propose the following conjecture. Conjecture 5.7.1. For any given integer l ≥ 3 and any integer a ≥ 3, there is a positive integer N(a) such that R(l, s + t − a) ≥ R(l, s) + R(l, t) − 1 for any s ≥ t ≥ N(a). Furthermore, there is an (l, s+ t− a)-graph G of order R(l, s)+ R(l, t)−2 such that V(G) = V1 ⋃ V2 , where G[V1 ] and G[V2 ] are an (l, s)-graph and an (l, t)-graph, respectively. We may propose the following conjecture similarly, which may be more difficult to prove. Conjecture 5.7.2. For any given integer l ≥ 3, there is positive ϵ(< 1/2), if a = ⌊ϵs⌋, then there is a positive integer N(ϵ), such that R(l, 2s − a) ≥ 2R(l, s) − 1 for any s ≥ N(ϵ).

46 | 5 Bi-color off-diagonal classical Ramsey numbers Furthermore, there is an (l, 2s − a)-graph G such that V(G) = V1 ⋃ V2 , where G[V1 ] and G[V2 ] are (l, s)-Ramsey graphs. If we add edges between two K3 -free graphs G1 and G2 by the triangle-free process, what can we say about the independence number of graph G obtained? Maybe for the graph G obtained this way, α(G) equals α(G1 ) + α(G2 ) or a little smaller, and is much larger than the independence number of the graph of order |V(G1 )| + |V(G2 )| obtained by the triangle-free process. That is to say, by the greedy approach we often obtain constructions that are not very good.

5.8 Problems of algorithms on off-diagonal Ramsey numbers It may be very difficult to determine the order of R(3, k). The main aim of this section is to conjecture a concrete “c” for which we can conjecture that R(3, k) ∼

ck2 ln k

by doing some computation. We conjecture that such a constant c exists. In this section, we will discuss the triangle-free process and its application in other graph classes, and related effect analysis. The triangle-free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. It is well known that R(s, k + l + 1) ≥ R(s, k + 1) + R(s, l + 1) − 1 . What about R(s, k + l) ≥ R(s, k + 1) + R(s, l + 1) − 1? Maybe it holds for all but a few cases, but we cannot prove it now. Conjecture 5.8.1. The inequality R(s, k + l) ≥ R(s, k + 1) + R(s, l + 1) − 1 holds for all but finite cases. It will be still difficult to prove there is a positive constant c such that R(s, k + l) ≥ R(s, k + 1) + R(s, l + 1) − c . For s = 3, it is not difficult to prove that R(3, k + l) ≥ R(3, k + 1) + R(3, l + 1) − 3, but this is weak. It is proved in a constructive way as follows. More similar constructions will be discussed later and can be found in [318]. Suppose that G is a (3, k + 1)-Ramsey graph and H is a (3, l + 1)-Ramsey graph. We delete a vertex u in G and a vertex v in H, and add edges with one end among the

5.8 Problems of algorithms on off-diagonal Ramsey numbers

| 47

neighbors of u in G and the other end among the neighbors of v in H. It is not difficult to prove that this newly constructed graph works. For instance, we can construct a (3, 4)Ramsey graph based on two copies of C5 this way. Note that C5 is a (3, 3)-Ramsey graph. Suppose that n is an integer and n > 50. Let us consider cyclic graphs generated in a way similar to the triangle-free process. That is to say, for any given integer n > 50, we generate a K3 -free cyclic graph randomly in the following way. We begin with an empty parameter set and add a new parameter to the parameter set with the same probability, subject to the constraint that no triangle is formed in the cyclic graph generated. We may obtain a (3, k)-cyclic graph, in which k is a given integer that is not very small. We wish that the independence number of the graph generated be small relative to the order n. That is to say, we wish that the lower bound for R(3, k) obtained this way not be too weak. Maybe we can generate quasi-random cyclic K3 -free graph based on a Sidon set containing 0. A set {a i | 1 ≤ i ≤ k} of integers is a Sidon set if the sums a i + a j , for i ≤ j are all different. That is, we can begin with a Sidon set A containing 0 in the parameter set and then add more parameters according to the suggestions above. Note that such a graph is K3 -free because A − {0} is sum-free for Sidon set A. Our general problem is, the graphs generated randomly in a smaller range, with cyclic graphs as examples, and those generated by the idea of triangle-free process that need not be cyclic graphs, which will be better in studying the lower bound for R(3, k)? Here “better” means to obtain a large lower bound on R(3, k) with a high probability by generating only a few graphs. We consider only the relation between the orders and the independence numbers of the K3 -free graphs generated and do not consider their difficulty in computing the independence numbers. Another idea is that we generate some (3, k 0 )-graphs of order n randomly by the triangle-free process and choose the graph with the minimum independence number among them and let it be G. Suppose that α(G) = k. Therefore, k < k 0 . Now let us construct a (3, t)-graph H of order 2n, such that in which t is small. We begin with two vertex disjoint subgraphs H1 and H2 of H, both of which are isomorphic to G; there are no edges between them at the beginning. Then we add edges between them following the triangle-free process, until the graph H obtained is edge maximal K3 -free. In the research based on this idea, one of the difficulties in the computation lies in computing the independence number of the graph obtained. To be feasible, in most cases we may suppose that n ≤ 100, because computing the independence number of a K3 -free graph of order no larger than 200 often does not need much computing time. Now let us discuss some computations that we have done. We can generate a graph of order 200 following the method discussed above. Let it be G1 , and the subgraph of G1 induced by the first 100 vertices be G2 , the subgraph induced by the second 100 vertices be G3 . Then we deal with G2 and G3 , respectively, by adding edges following the triangle-free process. Let the maximal K3 -free graphs obtained be G4

48 | 5 Bi-color off-diagonal classical Ramsey numbers

and G5 , respectively. Then we add edges between G4 and G5 with no edge between them at the beginning and let the maximal K3 -free graph obtained be G6 . Note that G i is K3 -free for any i ∈ {1, . . . , 6}. In the first experiment, the independence numbers of {G i | 1 ≤ i ≤ 6} are 36, 27, 27, 26, 25, and 38, respectively. In the second experiment, the independence numbers of these graphs are 36, 28, 28, 26, 24, and 41, respectively. It is necessary to do more similar computations and do some computations for other orders too. For instance, let us consider a known (3, 22)-cyclic graph H of order 130, of which the independence number is 21. If we use the method of the triangle-free process to add edges between two graphs without a common vertex that are both isomorphic to H with no edge between them at the beginning, to obtain a maximal K3 -free graph of order 260, maybe the independence number of the generated graph equals or is only a little smaller than 42. We wish that α(H) ≤ 37 hold for the graph of order 260 obtained, because the best known lower bound for R(3, 38) is 259. However, in the first graph of order 260 generated this way, we found an independent set of order 41 in only about 10 min. Note that computing the exact value of the independence number of a K3 -free graph of order 260 is often not easy. This makes us believe that for two graphs with small independence numbers, the independence number of the large graph obtained based on the triangle-free process may rarely be much smaller than sum of the independence numbers of the two small graphs. More computations will be necessary. It is interesting to generate a K3 -free graph G of order n, such that α(G) is smaller than that of the graphs generated by the triangle-free process. For instance, in order to bound α(G), we may control the maximum degree. We may also bound minimum degrees of the graphs generated, which may make the independence numbers a little smaller with a higher probability. Another idea is that when we add edges, we may find large independent sets and make them smaller by adding edges between a pair of vertices in them. Here, it is enough to search large independent sets by a fast approximation algorithm instead of by an exact one. It is not easy to tell if we can obtain interesting lower bound on R(3, k) based on these ideas before we do more computations and a more detailed analysis. If we bound the degrees at the beginning, we may obtain better constructions with a higher probability. Otherwise, if the maximum degrees of the graphs generated are too large, then the independence numbers must be large too, which makes the graphs of little interest in studying the lower bound on R(3, k) for small k. Can we improve the known lower bound on r(4, 21), 242, by a similar idea? What we are suggesting is to add edges between two subgraphs isomorphic to G127 , to construct a K4 -free graph with independence number no larger than 20. Here, G127 is the known (4, 12)-graph defined by the cubic residues of 127; G127 is of strong symmetry. However, this is of little use in the computation because the edges added randomly are so many that the large graphs obtained are not of global high symmetry.

5.9 Constructive lower bounds for off-diagonal Ramsey numbers |

49

There are so many ways to try to do this, and we need to compute the lower bounds on the independence numbers of many graphs. So, we need a fast algorithm for lower bounds on the independence numbers in these computations. The following problems may be interesting for some readers. All of these problems may be difficult, and we will not discuss them in detail. It is interesting to analyze and conjecture the details of the known method on the upper bound for R(3, k), which is related to the greedy algorithm on computing independence numbers of K3 -free graphs. More details can be found in [42]. We may also generate a (3, k)-graph randomly as a subgraph of a Turán graph T n,r , where the order of T n,r is n and α(T n,r ) = ⌈n/r⌉. To make the graph generated be a (3, k)-graph with a high probability, here n should equal the best known lower bound, and r should be chosen so that ⌈n/r⌉ is smaller than k, and k − ⌈n/r⌉ should not be too small. We should consider small R(3, k) for which the gap between the best known lower and upper bounds is not small.

5.9 Constructive lower bounds for off-diagonal Ramsey numbers The best known lower bounds for many Ramsey numbers are not constructive. They were obtained by non-constructive methods. That is, the problem of construction is not answered. In most cases, it is not feasible to obtain a constructive lower bound of the same or similar order by adjusting the non-constructive method. For Ramsey numbers and other similar topics in combinatorics, how to obtain a good construction for which the existence was proved by non-constructive methods is a general problem that is both difficult and important. In fact, we are not sure if we can obtain constructive results matching the non-constructive ones in the future, or if it is impossible for some problems. The best known explicit constructive lower bound for R(3, k) was given in [6], where it was proved that R(3, k) ≥ Ω(k3/2 ) constructively. In [174], explicit constructive lower bounds for more bi-color off-diagonal Ramsey numbers were obtained. These new results are R(4, k) ≥ Ω(k 8/5 ), R(5, k) ≥ Ω(k 5/3 ) and R(6, k) ≥ Ω(k 2 ). For the constructive lower bounds in [6] and [174] cited above, we should write the following forms with more information than above. We take R(3, k) as an example, as follows. What was proved in [6] is the following inequality: R(3, 2(q2 + q + 1) + 1) > (q + 1)(q2 + q + 1) , where q is a prime power. These constructive methods give weak lower bounds for small Ramsey numbers. For instance, if q = 3, then we obtain R(3, 27) ≥ 53 by the inequality above, which is much smaller than the lower bound 101 that we can easily obtain by the triangle-free process, let alone the best known lower bound 167 obtained in [297]. Similarly, for q = 4 or q = 5, we can only obtain R(3, 43) > 105 and R(3, 63) > 186, respectively, by the inequality above.

50 | 5 Bi-color off-diagonal classical Ramsey numbers

A natural question we may ask here is, how large can the constructive lower bound for R(7, k) be? What about the lower bound on R(s0 , k) for given integer s0 no smaller than 7? Because all these constructive lower bounds are weaker than the best known ones, we propose the following problem. Problem 5.9.1. For given integer s ≥ 3, improve the known constructive lower bound for R(s, k). In fact, we do not know if we can improve these known constructive lower bounds. Maybe it is difficult to give a good constructive lower bound for R(s, k), and the better the lower bound, the more complex the construction, and we will never give a good constructive general lower bound in finite words that matches the best known nonconstructive lower bound.

5.10 Ramsey graphs In this section, let us discuss some (s, t)-Ramsey graphs. Based on Theorem 3.1.1 in Chapter 3, the connectivity of the (k, l)- Ramsey graph was studied in the Master’s thesis of Xiaodong Xu in 2002. Xu proved that for any integers k, l ≥ 3, the connectivity of any (k, l)-Ramsey graph is no smaller than k − 2. The Master’s thesis of Xiaodong Xu was written in Chinese, and few people have read it. In a talk at the 2005 British Combinatorial Conference, David Penman conjectured that for k ≥ 3 every (k, k)-Ramsey graph is connected. This had been proved in the Master’s thesis of Xiaodong Xu, but those mathematicians were not able to read it and did not know this result at all. Beveridge and Pikhurko did better in [31]. By using Theorem 3.1.1 in Chapter 3, they proved the following theorem. Theorem 5.10.1. For any k, l ≥ 3, the connectivity of any (k, l)-Ramsey graph is no less than k − 1. Beveridge and Pikhurko also studied the edge connectivity of Ramsey graphs in [31] and proved the following theorem. Theorem 5.10.2. Let k, l ≥ 3, and G be a (k, l)-Ramsey graph. The edge connectivity of G satisfies λ(G) ≥ min{δ(G), κ(G) + k − 3} . In [309], the lower bound on connectivity of the Ramsey graph in Theorem 5.10.1 was increased to k for k ≥ 5. Similarly, the following Theorem 5.10.4 improves a result in [31] on Ramsey graphs that are Hamiltonian. These results directly depend on Theorem 3.1.2 in Chapter 3, in such a way that they could be further strengthened if the lower bound in Theorem 3.1.2 in Chapter 3 were improved. Now, let us cite the following theorems from [309].

5.10 Ramsey graphs |

51

Theorem 5.10.3. If k ≥ 5 and l ≥ 3, then the connectivity of any (k, l)-Ramsey graph is no less than k. Theorem 5.10.4. If k ≥ l −1 ≥ 1 and k ≥ 3, except (k, l) = (3, 2), then any (k, l)-Ramsey graph is Hamiltonian. Here, the following theorem of Chvátal and Erdős proved in [65] on Hamilton graphs was used. Theorem 5.10.5. Let G be a graph with at least three vertices. If, for some s, G is sconnected and contains no independent set of more than s vertices, then G has a Hamiltonian circuit. In particular, for k ≥ 3, all diagonal (k, k)-Ramsey graphs are Hamiltonian. It is an interesting open question for which k and l, all (k, l)-Ramsey graphs are Hamiltonian, where 3 ≤ k < l − 1. The authors of [309] expected it to be true at least when k is sufficiently close to l. Let us propose the following two conjectures on Ramsey graphs. Conjecture 5.10.1. For any integers s and t no smaller than 3, there is an (s, t)-Ramsey graph that is Hamiltonian. Conjecture 5.10.2. For any integers s and t no smaller than 3, there is an (s, t)-Ramsey graph of which the connectivity equals to the minimum degree. To study the properties of Ramsey graphs without knowing the exact values of Ramsey numbers, or the construction of a Ramsey graph, is a little similar to the qualitative theory of ordinary differential equations. It seems to have been impossible to study the connectivity of a Ramsey graph before those methods including Theorem 3.1.1 in Chapter 3 in [313] appeared. Let us cite two more theorems from [65]. Theorem 5.10.6. Let G be an s-connected graph with no independent set of more than s + 2 vertices. Then G has a Hamiltonian path. Theorem 5.10.7. Let G be an s-connected graph with no independent set of more than s vertices. Then G is Hamiltonian-connected (i.e., every pair of vertices is joined by a Hamiltonian path). By these theorems it is not difficult to obtain related results on Ramsey graphs. Let us consider only the following theorem based on Theorem 5.10.6. Theorem 5.10.8. Suppose s ≥ 5 and t ≤ s + 2. If G is an (s, t)-Ramsey graph, then G has a Hamiltonian path. By Theorem 5.10.4 we know that when s ≥ 5 and t ≤ s + 1, any (s, t)-Ramsey graph has a Hamiltonian cycle, and then has a Hamiltonian path too. Therefore, the new case in Theorem 5.10.8 is that every (s, s + 2)-Ramsey graph contains a Hamiltonian path

52 | 5 Bi-color off-diagonal classical Ramsey numbers for any s ≥ 5. We hope that mathematicians doing research on Hamiltonian graphs will find the problems in this section interesting. Let us discuss more topics in the following two subsections.

5.10.1 Disjoint k − 1-cliques in (k, l)-Ramsey graphs In [316], some inequalities on classical Ramsey numbers were proved. Most of the results are similar to those proved by Abbott based on G[H]. However, the following theorem on Ramsey graphs proved in [316] is new. Theorem 5.10.9. Let G be a (k, l)-Ramsey graph. Then there exists k − 1 disjoint l − 1independent sets in G, and there exists l − 1 disjoint k − 1-cliques in G. This theorem is simple but interesting. It seems it would not be easy to improve, although it may be weak, at least for cases in which both k and l are large. If we can find a method to improve it, then we may understand classical Ramsey numbers and Ramsey graphs better than before. Let us consider an example. Let G be a given (6, 7)-Ramsey graph. We know that R(6, 7) − R(6, 6) ≥ 10. If we find two disjoint 5-cliques in G, can the subgraph H obtained by deleting these two 5-cliques be a K5 -free graph? By the theorem cited above, we know that it cannot. Note that only by the order of H we do not know whether there is a 5-clique in H, because the order of H is R(6, 7) − 1 − 10, and we do not know if it is smaller than R(6, 6). If both k and l are not very small, there may be much more disjoint l − 1-independent sets and k − 1-cliques in a (k, l)-Ramsey graph. For instance, it seems impossible that there are not ten disjoint 9-cliques in a (10, 10)-Ramsey graphs. Our problem is as follows. Problem 5.10.1. For given integers k and l that are no smaller than 3, study the number of disjoint k − 1-cliques in (k, l)-Ramsey graphs.

5.10.2 Turán type theorems for k-connected graphs In [47], Turán type theorems for k-connected graphs were obtained, where Turán type theorems are in the form of a given stability number (i.e., independence number). Let f(n, α, k) denote the minimum size of a k-connected graph with order n and stability number α. The following conjecture on f(n, α, k) was proposed in [47], which may be used to study the relation between the connectivity and the minimum degree of the Ramsey graph. Conjecture 5.10.3. Let n, α, k be three integers such that n ≥ 2α, n ≥ α + k, α ≥ 2 and k ≥ 3. Then f(n, α, k) = ⌈nk/2⌉, if n ≤ kα and f(n, α, k) = t1 (n, α) + ⌈kα/2⌉

5.10 Ramsey graphs

| 53

otherwise, where t1 (n, α) is the minimum size of the graph that consists of α disjoint balanced cliques and is of order n and stability number α. Note that Conjecture 5.10.3 was proved to be true when α ≥ 3 and n ≥ ⌈(k − 2)α/2⌉ + 2α (see [47]). It is not difficult to see that f(n, α, k) ≤ t1 (n, α) + ⌈kα/2⌉ for n ≥ (k + 1)α.

6 Multicolor classical Ramsey numbers In this chapter we discuss multicolor classical Ramsey numbers. There are fewer known results on multicolored classical Ramsey numbers than on the bi-color ones, among which results on upper bounds are very few and most lower bounds seem weak. Known lower and upper bounds for some small multicolor classical Ramsey numbers can be found in [229]. People have known the exact value of R(3, 3, 3) since 1955 (see [151]), for which the upper bound is easy because R(3, 3, 3) ≤ 2 + 3(R(3, 3) − 1) = 17, and the lower bound is not difficult to prove too. The second non-trivial multicolor classical Ramsey number that we know the exact value is R(3, 3, 4) = 30, of which the lower bound had been known for some years, and the upper bound was obtained by detailed analysis and much computation [66]. We often use G[H] in studying the lower bound for multicolor classical Ramsey numbers and some multicolor generalized Ramsey numbers. In many cases, the lower bounds obtained this way seem weak. However, we do not know how to do much better now.

6.1 Results obtained by methods used on R(s, t) Some know methods used in studying bi-color classical Ramsey numbers, may be used on the multicolor ones too. For instance, similar to R(s, t) ≤ R(s − 1, t) + R(s, t − 1) , in [151] Greenwood and Gleason proved that r

R(k 1 , . . . , k r ) ≤ 2 − r + ∑ R(k 1 , . . . , k i−1 , k i − 1, k i+1 , . . . , k r ) . i=1

Another example is the inequality R m+n (k) − 1 > (R m (k) − 1)(R n (k) − 1) can be proved based on the same idea with Theorem 3.1.1 in Chapter 3 that improves R(k, s + t − 1) ≥ R(k, s) + R(k, t) − 1. We can see that R m+n (k) − 1 > (R m (k) − 1)(R n (k) − 1) is much weaker than Theorem 6.2.2 in this chapter (proved in [306]), but improves R m+n (k) − 1 ≥ (R m (k) − 1)(R n (k) − 1) by one.

https://doi.org/10.1515/9783110576702-006

56 | 6 Multicolor classical Ramsey numbers

6.2 On the multicolor classical Ramsey number R n (k) In this section we will focus on the bounds for multicolor diagonal classical Ramsey numbers R n (k). Because R n (k) is related to the Shannon capacities of graphs with bounded independence numbers [106], we will write a subsection on the Shannon capacity of graphs before we discuss R n (3) and other multicolor classical Ramsey numbers.

6.2.1 The Shannon capacity of graphs Most of this section was cited from [306]. Shannon defined the Shannon capacity of a graph in [251]. For arbitrary graphs G1 , . . . , G n , where G i = (V i , E i ), the graph product G1 × ⋅ ⋅ ⋅ × G n is defined to be a graph G on the vertex set V = V1 ×⋅ ⋅ ⋅×V n , whose edges are all pairs of distinct vertices {(u 1 , . . . , u n ), (v1 , . . . , v n )}, such that for each i from 1 to n, u i = v i or {u i , v i } ∈ E i . This product is associative, and also commutative up to isomorphisms permuting the coordinates. G n denotes the n-fold product of the same graph G, namely G n = ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ G × ⋅ ⋅ ⋅ × G. The capacity c(G) of a graph G was defined by Shannon [251] as the n limit c(G) = lim α(G n )1/n n→∞

and is now called the Shannon capacity of a noisy channel modelled by graph G (see [10, 12]). The quantity c(G) is often simply referred to as the Shannon capacity of G. The study of c(G) within information theory was initiated by Shannon [251] and has grown to be an extensive area involving electrical engineering, communication theory, coding theory, and other fields that typically use probability theory as a tool. It may be less known that c(G) attracted attention of many graph theorists trying to compute it [10–12, 178, 244]. Suppose that we have a set Σ of k characters which we wish to send over a noisy channel one at a time. Let V(G) = Σ, and assume further that the edges of G indicate a possible confusion between pairs of characters when transmitted over the channel. When sending a single character, the maximum number of characters we can fix, and then choose from for transmission without danger of confusion, is α(G). When we use the same channel repeatedly n times, we could obviously send α(G)n words of length n by using an independent set in G at each coordinate. However, we might be able to do better by sending words from Σ n corresponding to vertices of an independent set of order α(G n ) in graph G n , in cases when the general inequality α(G n ) ≥ α(G)n is strict. The Shannon capacity c(G) measures the efficiency of the best possible strategy when sending long words over a noisy channel modelled by G, since the limit limn→∞ α(G n )1/n defining it can be seen as approaching the effective alphabet size in zero-error transmissions.

6.2 On the multicolor classical Ramsey number R n (k) | 57

Shannon asked the Shannon capacity of C5 , and obtained a lower bound √5 in [251]. Lovász determined it to be exact in [189]. The method in [189] is an important method in studying the upper bounds for Shannon capacities of graphs. In [189], an upper bound on the Shannon capacity of a graph which is called Lovász number, was introduced. There are many references on the Lovász number.

6.2.2 R n (3) and the Shannon capacity of graphs with bounded independence numbers The best known general lower bound for R n (3) was given based on a known inequality on Schur numbers. Schur numbers will be discussed in details in Section 12.3. The best known upper bound on R n (3) for n ≥ 4 was given in [311], which was obtained based on R4 (3) ≤ 62. This upper bound is R n (3) ≤ 1 + (e − 16 ) n! . Note that this best known upper bound for R n (3) is only a little better than the one given by Schur about one hundred years ago. It is well known that R m+n (k) − 1 ≥ (R m (k) − 1)(R n (k) − 1) was proved in [1]. We can see that the limit limn→∞ [R n (k)]1/n exists for any integer k ≥ 3 (see [58]), and may equal to ∞. The following Conjecture 6.2.1 is a very old problem of Erdős’, and it is one of the most famous conjectures in Ramsey theory. It is related to the Shannon capacity of graphs with independence numbers 2 (see [106, 251]). Conjecture 6.2.1. The limit limn→∞ [R n (3)]1/n = ∞. It is obvious that Conjecture 6.2.1 implies the following conjecture. Conjecture 6.2.2. For given integer k ≥ 3, limn→∞ [R n (k)]1/n = ∞. The following conjecture seems a little easier, if it is true. Of course, it seems far from reach now, and may be almost as difficult as Conjecture 6.2.1. Conjecture 6.2.3. There is a positive integer k such that 1

lim [R n (k)] n = ∞ .

n→∞

It will be a little surprising if limn→∞ [R n (k 0 )]1/n = ∞ and limn→∞ [R n (k 0 − 1)]1/n is finite for some integer k 0 ≥ 4. It is not difficult to see that Conjecture 6.2.1 holds if the following conjecture holds. Conjecture 6.2.4. lim

n→∞

R n+1 (3) =∞. R n (3)

58 | 6 Multicolor classical Ramsey numbers We may consider if there is an integer k ≥ 3 such that limn→∞ (R n+1 (k))/(R n (k)) = ∞ first. Note that in almost all cases, what we know about different Ramsey numbers are weak. Xiaodong Xu and Radziszowski have studied Conjecture 6.2.3 for more than one decade, and found that this conjecture seems very far from our reach. We have little idea how to prove the following inequalities. Does R n (3) > (a log n)n hold for some positive constant a? Does R n (3) < (n/b)n hold for some positive constant b? We can see that if R n (3) > (a log n)n holds, then Conjecture 6.2.1 holds too. In fact, we do not know if (R n (3))1/n > 4 holds for n large enough, and the best known lower bound for limn→∞ [R n (3)]1/n is 3211/5 . Suppose that m ≥ 5. We know that R m (3) ≤ m(R m−1 (3) − 1) + 2. It is interesting to consider the following problem. Problem 6.2.1. Suppose that m ≥ 5. Is there a good coloring for the lower bound on R m (3), such that there is a vertex v of degree R m−1 (3) − 1 in the monochromatic subgraph induce by the edges in color i for any i ∈ {1, 2, 3, 4}? If there is such a good coloring, then we obtain a lower bound 4R m−1 (3) − 2 for R m (3). In fact, we can do a little better to obtain R m (3) ≥ 4R m−1 (3) − 2 + R m−4 (3) − 2 = 4R m−1 (3) + R m−4 (3) − 4 for m ≥ 5, where we define R1 (k) = k. Let us consider the details now. Suppose that V(G) = {v, v1 , . . . , v x }, where x = 4(R m−1 (3) − 1), and v is the vertex v in Problem 6.2.1, we can add more R m−4 (3) − 2 vertices {u 1 , . . . , u y }, where y = R m−4 (3) − 2, such that the color of the edge (u i , v j ) is same to the color of edge (v, v j ) for any i ∈ {1, . . . , y} and j ∈ {1, . . . , x}. Then we can color all the edges in the subgraph induced by {v, u 1 , . . . , u y } with colors in {color 5, . . . , color m} such that it become a Ramsey-coloring for R m−4 (3). We can see the graph constructed is a good coloring for R m (3), of which the order is 4(R m−1 (3) − 1) + 1 + R m−4 (3) − 2 = 4R m−1 (3) + R m−4 (3) − 5. Hence we have R m (3) ≥ 4R m−1 (3) + R m−4 (3) − 4 if such a good coloring for the lower bound on R m (3) in Problem 6.2.1 exists. We do not know if such a good coloring exists even for R5 (3). Note that the known lower and upper bounds for R5 (3) are 162 ≤ R5 (3) ≤ 307. If the answer for Problem 6.2.1 is yes, then we can improve the lower bound for R5 (3) to 4 × 51 + 5 − 4 = 205, what may be too nice to be true. Maybe such a good coloring in Problem 6.2.1 exists when m is large enough. It may be interesting for us to consider the history of G[H], with applications on the lower bounds for multicolor Ramsey numbers and the upper bounds for Folkman numbers. If the edge relations of the two graphs are order relations, then the edge relation of their lexicographic product is the corresponding lexicographic order. The lexicographic product was first studied by Felix Hausdorff in [162]. The problem of recogniz-

6.2 On the multicolor classical Ramsey number R n (k) |

59

ing whether a graph is a lexicographic product is equivalent in complexity to the graph isomorphism problem. This is not obvious, and Feigenbaum and Schäffer showed it to be true in [119]. Some simple results on the chromatic number and other functions of the lexicographic product were proved in [131], among which some must be known by Abbott no later than 1965 (see [1]). We believe this because Abbott proved some results on the lower bounds for classical Ramsey numbers based on the lexicographic product of graphs, i.e., the composition of graphs. A result in [312] based on G[H] was given as follows. It is a simple generalization of the diagonal case. Theorem 6.2.1. If k j ≥ 2 for 1 ≤ j ≤ r, then for all i = 1, . . . , r − 1 R(k 1 , . . . , k r ) > (R(k 1 , . . . , k i ) − 1)(R(k i+1 , . . . , k r ) − 1).

(1)

In [306], the inequality R2n (k)−1 ≥ (R n (k)−1)2 was improved into the following Theorem 6.2.2. Theorem 6.2.2 can be generalized to the off-diagonal case without difficulty. Note that this inequality cannot be proved by probabilistic methods. Theorem 6.2.2. For integers k, n, m, s ≥ 2, let G ∈ Rn (k; s) be a coloring containing an induced subcoloring of K m using less than n colors. Then k, . . . , k ) − 1) + 1 . R2n (k) ≥ s2 + m(R n (k − 1, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

(2)

n−1

In [58] Chung proved that R a (3) ≥ 3R a−1 (3) + R a−3 (3) − 3 . This lower bound was proved by a constructive method. In [240], Aaron Robertson generalized this inequality into the off-diagonal cases, i.e., R(3, 3, 3, k 1 , k 2 , . . . , k r ) ≥ 3R(3, 3, k1 , k 2 , . . . , k r ) + R(k 1 , k 2 , . . . , k r ) − 3 holds for any positive r and for any k i ≥ 3, i = 1, 2, . . . , r. In [312], the following theorem was proved. Theorem 6.2.3. Let G ∈ R(k 1 , . . . , k r ; 2n + 1), for k i ≥ 3, 1 ≤ i ≤ r, and suppose that for some partition V(G) = V1 ⋃ V2 ⋃{w} the induced subcolorings G[V1 ] and G[V2 ] are isomorphic. Then, given any (3, s1 , . . . , s t ; m)-coloring with a vertex of degree d in color 1, we have constructively R(s1 , . . . , s t , k 1 , . . . , k r ) ≥ mn + d + 1 . It seems impossible to prove Conjecture 6.2.2 by considering the method used in the proof of Theorem 6.2.3 in more details. We may propose the following problem on the bi-color classical Ramsey number which is related to the case r = 2 in Theorem 6.2.3.

60 | 6 Multicolor classical Ramsey numbers

Given positive integers s and t, let the twin Ramsey number n(s, t) be the maximum integer n such that there is an (s, t)-graph G of order n, where V(G) = V1 ⋃ V2 , V1 ⋂ V2 = Ø, and G[V1 ] is isomorphic to G[V2 ]. It is not difficult to know that n(s, t) = n(t, s) and n(2, t) = 2⌊t/2⌋ for any s, t ≥ 3. We can see that n(s, t) < R(s, t). We may conjecture that n(s, t) is small at first, but note that n(s, t) ≥ H(s, t) − 1, and n(s, t) ≥ H(s, t) when H(s, t) is even, where H(s, t) is the order of the maximum cyclic (s, t)-graph. On the other hand, we have no idea on the difference between R(s, t) and H(s, t). People have obtained many results on H(s, t) in studying lower bounds for small Ramsey numbers, for instance, [161]. We may consider the lower bound for n(s, t) based on Cayley graphs that is not cyclic. The following theorem was proved in [312]. Theorem 6.2.4. For k i ≥ 3, 1 ≤ i ≤ r, s, t ≥ 3, and given any (3, k 1 , . . . , k r ; m)coloring, we have constructively R(k 1 , . . . , k r , s, t) ≥ 12 mn(s, t) + 1 . We may generalized n(s, t) to the multicolor case n(k 1 , . . . , k r ) without difficulty. Let n(k 1 , . . . , k r ) be the maximum integer such that there is an (k 1 , . . . , k r )-graph of G of order n, where V(G) = V1 ⋃ V2 , V1 ∩ V2 = Ø, and G[V1 ] is isomorphic to G[V2 ]. Now, let us discuss the Shannon capacity of graphs with bounded independence numbers. In [306] it was proved that the construction obtained implied that the supremum of the Shannon capacity over all graphs with independence number 2 cannot be achieved by any finite graph power, and this was generalized to graphs with bounded independence numbers. On the other hand, the following conjecture seems interesting. Conjecture 6.2.5. For given integer k ≥ 3, there is not a finite graph G with independence number k − 1, such that the Shannon capacity of G equals to lim n→∞ [R n (k)]1/n . If Conjecture 6.2.5 is false, then limn→∞ [R n (k)]1/n is finite and limn→∞ [R n (k)]1/n ≤ |V(G)|. So Conjecture 6.2.2 implies Conjecture 6.2.5. Based on the results in [306], we can see that if there is such a finite graph G, then the Shannon capacity of G cannot be achieved by (α(G m ))1/m for any positive integer m. It is not difficult for the readers of [106] to prove that Conjecture 6.2.5 is not true for infinite graph. Such a result was not proved in [106], but we may prove it based on the same method in it without difficulty. We know that there is a graph G such that α(G) = k − 1 and α(G m ) = R m (k) − 1 by the result in [106]. It is interesting to consider the minimum order of such a graph G. We know that |V(G)| can be as small as m(R m (k)− 1). We can obtain a graph of smaller order by constructing universal graphs.

6.2 On the multicolor classical Ramsey number R n (k) |

61

Problem 6.2.2. Study the minimum order of graph G such that α(G) = k − 1 and α(G m ) = R m (k) − 1. The following problem seems interesting. Problem 6.2.3. Is there an integer i0 ≥ 3 such that [R i (k) − 1]1/i monotone increasing for i ≥ i0 and given integer k? We have done some computation based on the known lower bounds for small diagonal multicolor Ramsey numbers R i (k), where k ∈ {3, 4, 5}. We know that R3 (4) ≥ 128 and R4 (4) ≥ 634. We can see that 1271/3 > 6331/4 . Maybe R3 (4) = 128 and R4 (4) > 634, and this is not a counter-example for Problem 6.2.3. We also know that R3 (5) ≥ 417 and R4 (5) ≥ 3049. We can see that 4161/3 > 30481/4 . Maybe this is because the known lower bound for R4 (5), 3049, is weak and much smaller than the exact value. Let us consider another problem based on the results in [106]. In [232] it was proved that if R4 (3) = 51 then R4 (K3 + e) = 52, otherwise R4 (K3 + e) = R4 (3). The following conjecture on the value of R4 (3) is proposed by Radziszowski. Conjecture 6.2.6. R4 (3) = 51.

6.2.3 More problems on R n (k) We know that R2k (3) − 1 ≥ (R k (3) − 1)2 and R k (5) − 1 ≥ (R k (3) − 1)2 . We know better inequalities, but the ones we cite here seem enough to our analysis. Because limk→∞ [R2k (3)]1/(2k) = limk→∞ [R k (3)]1/k , we have limk→∞ [R2k (3)]1/k = (limk→∞ [R k (3)]1/k )2 . Maybe both sides in this equality equal to ∞. By R k (5) − 1 ≥ (R k (3) − 1)2 we know that 1

1

2

1

lim [R k (5)] k ≥ ( lim [R k (3)] k ) = lim [R2k (3)] k .

k→∞

k→∞

k→∞

Hence limk→∞ [R k (5)]1/k ≥ limk→∞ [R2k (3)]1/k . It is to say, limk→∞ ([R k (5)]1/k )/ ([R2k (3)]1/k ) ≥ 1. Note that this limit exists, and may be 1 or ∞, or a finite value larger than 1. What we like to ask is: does the inequality above hold strictly? Problem 6.2.4. limk→∞ (R k (5)/R2k (3))1/k > 1? We know that R(5, 5) ≤ 49 < 51 ≤ R4 (3). Therefore R(5, 5) < R4 (3). However, for any two Ramsey numbers among R6 (3), R(5, 5, 5), R(3, 3, 5, 5), R(3, 3, 3, 3, 5), we do not know which is larger and which is smaller in the cases of any pairs. Known lower and upper bounds for these classical Ramsey numbers may be weak.

62 | 6 Multicolor classical Ramsey numbers For any integer k ≥ 3, the best known upper bound for R k (5) is larger than that for R2k (3). The best known upper bound here means the upper bound that we can obtain by computing without difficulty. Note that if s, t, k ≥ 2 and s ≥ t, then R k (st + 1) ≥ R k (s + 1)(R k (t + 1) − 1) was proved in [312]. Therefore we know that R k (5) ≥ R k (3)(R k (3) − 1) . We also know the following lower bound for R2k (3) proved in [306], R2k (3) ≥ (R k (3) − 1)2 + ⌈

R k (3) − 2 ⌉ (R k (3) − 1) + 1 . n

We can see that R k (3)(R k (3) − 1) < (R k (3) − 1)2 + ⌈

R k (3) − 2 ⌉ (R k (3) − 1) + 1 . n

It is to say, the known lower bound for R k (5) cited above is smaller than the lower bound for R2k (3) proved in [306]. Furthermore, we know that Chung proved that R k (3) ≥ 3R k−1 (3) + R k−3 (3) − 3 in [58], and we have cited this inequality earlier. By this inequality we can obtain 9R2k−2 (3) < R2k (3). It will be interesting to prove an inequality on R k (5) similar to R k (3) ≥ 3R k−1 (3) + R k−3 (3) − 3. However, we cannot prove R k (5) > 5R k−1 (5) now. We list it as a problem here, and wish that it is not difficult for some mathematicians. Problem 6.2.5. Does R k (5) > 5R k−1 (5) hold for every integer k ≥ 3? Let L(k 1 , . . . , k r ) denote the maximum order of any cyclic (k 1 , . . . , k r )-coloring. Therefore R(k 1 , . . . , k r ) ≥ L(k 1 , . . . , k r ) + 1. In 1968, Giraud proved the following theorem in [137](see also [312]), by which we can obtain a lower bound c ⋅ 7k for R k (5). Theorem 6.2.5. For k i ≥ 3, i = 1, . . . , r, L(k 1 , . . . , k r , k r+1 ) ≥ (2k r+1 − 3)L(k 1 , . . . , k r ) − k r+1 + 2 . Let us consider the following conjecture. Conjecture 6.2.7. If s1 , . . . , s r is a monotonically increasing integer sequence, and s1 ≥ 2, then R(s1 , . . . , s r ) ≥ R(s1 , . . . , s i − 1, . . . , s j + 1, . . . , s r ) . Wang Rui believes this conjecture to be true, but his proof in [293] is not correct, and this conjecture is still open now. We may consider R k (10) − 1 > (R k (4) − 1)2 . Although we can discuss related problems on Shannon capacity, we omit the details.

6.2 On the multicolor classical Ramsey number R n (k) | 63

In [312], it was proved that if k 1 ≥ 5, then R(k 1 , 2k 2 − 1, . . . , k r ) ≥ 4R(k 1 − 1, k 2 , . . . , k r ) − 3 . Let limk→∞ [R k (i)]1/k = a i . If a5 is finite, then by R(5, 5, k 1 , . . . , k r ) ≥ 4R(3, 4, k1 , . . . , k r ) − 3 , and for t ≥ 2, we have R2t (5) − 1 ≥ 4t (R t (3) − 1)(R t (4) − 1) , therefore a25 ≥ 4a3 a4 , which may be weak, but we do not know how to improve it. Suppose that limk→∞ ((R k (a + 1))/(R k (a)))1/k = f(a). We can see that the limit f(a) always exists. It is easy to see that f(a) cannot equal to 1 for all a ≥ 3. Let us propose the following problem. Problem 6.2.6. Does lim k→∞ ((R k (a + 1))/(R k (a)))1/k > 1 hold for every integer a ≥ 3? Maybe f(a) always exists and equals to ∞ for any a ≥ 3. We may study f(3) as an example. It is also interesting to know if limk→∞ (R k (a + 1))/(R k (a)) = ∞. We do not know if limk→∞ (R k (4))/(R k (3)) = ∞. If limk→∞ (R k (a + 1))/(R k (a)) is finite, then limk→∞ ((R k (a + 1))/(R k (a)))1/k = 1. It is to say, if limk→∞ (R k (a + 1))/(R k (a)) is finite, then limk→∞ (R k (a + 1))1/k = limk→∞ (R k (a))1/k . We like to know the answer of the following problem. Note that we know nothing on the exact values of multicolored diagonal classical Ramsey numbers other than R(3, 3, 3) = 17, and it is difficult to do interesting computing on these problems. Problem 6.2.7. Does (R i+1 (a))/(R i (a)) ≥ (R i (a))/(R i−1 (a)) hold for all a ≥ 3 and i ≥ 3? Does limn→∞ (R(k, k, k) − 1)/((k − 1)(R(k, k) − 1)) exist and equal to ∞? Of course, if it exists, it equals to limn→∞ (R(k, k, k))/(kR(k, k)). We can prove that R(k, k, k) > (R(k, k, 3) − 1)⌊(k − 1)/2⌋ similar to Theorem 6.2.4. However, we do not know if (R(k, k, 3))/(R(k, k)) tends to ∞ when k tends to ∞. We can see that if limk→∞ (R(k, k, 3))/(R(k, k)) = ∞, then limn→∞ (R(k, k, k) − 1)/ ((k − 1)(R(k, k) − 1)) = ∞. Now we know only that lim k→∞ (R(k, k, 3))/(R(k, k)) ≥ 5/2, if it exists, based on a result in [312]. This seems weak. If the method based on random graph theory used in studying the upper bound on edge Folkman number in [243], can be used in studying lower bounds for classical Ramsey numbers, is an interesting problem. Even if it can, it may be used through very different analysis and computation. Does R k (3) ≥ R(k, k) hold for any integer k ≥ 3? In 2001, Xiaodong Xu proposed this problem, but no idea has been found until today. It is not difficult to know that it holds for any positive integer k ≤ 10.

64 | 6 Multicolor classical Ramsey numbers It is not difficult to see that if there are infinite cases in which R k (3) < R(k, k) holds, then limn→∞ [R n (3)]1/n ≤ 4. Maybe R k (3) ≥ R(k, k) hold for any integer k ≥ 3. Let us propose the following conjecture here. Conjecture 6.2.8. There are at most finite cases in which R k (3) < R(k, k) holds. Let us consider the bounds for some small multicolor classical Ramsey numbers. We know that 128 ≤ R(4, 4, 4) ≤ 236. Both bounds have not been improved since 1982 when the lower bound was obtained, until recently R(3, 3, 4) = 30 was proved in [66]. We can see that R(3, 4, 4) ≤ R(3, 3, 4)+R(3, 3, 4)+R(2, 4, 4)−3+2 = 30+30+18−1 = 77, and R(4, 4, 4) ≤ 3(R(3, 4, 4) − 1) + 2 ≤ 3 × 76 + 2 = 230 follows. It seems that the only possible way to improve R(4, 4, 4) ≤ 230 is to improve R(3, 4, 4) ≤ 77, which seems difficult. Boza, Dybizbański, and Dzido [51] reviewed that probably R(4, 4, 4) can never be solved. We conjecture that the best known upper bound for R(4, 4, 4) is much weaker than the lower one. We conjecture that the R(4, 4, 4) ≤ (1/2)(128 + 230) = 179. It is to say, we conjecture that R(4, 4, 4) is no larger than the mean number of the known lower and upper bounds. Maybe R(4, 4, 4) is much smaller. Conjecture 6.2.9. R(4, 4, 4) ≤ 179. By the way, we have R(3, 4, 5) ≤ R(2, 4, 5) + R(3, 3, 5) + R(3, 4, 4) − 1 ≤ 25 + 57 + 77 − 1 = 158 . In general, known lower bounds for multicolor classical Ramsey numbers are weak, but known upper bound may be much weaker.

6.3 Other multicolor classical Ramsey numbers For two graphs H, K and a positive integer k, let r k (H; K) denote the Ramsey number R(H1 , . . . , H k , K), where H i = H for all i ∈ {1, . . . , k}. In [14] it was shown that for every fixed positive integer k, Ω(

m k+1 (log log m)k−1 m k+1 ) ≤ r k (K3 ; K m ) ≤ c k . 2k+δ (log m) (log m)k

The lower bound was obtained by using the probabilistic method and constructive results together, and itself is not constructive. One of the interesting case included in this result is R(3, 3, m). It seems that these methods and results are difficult to be used in studying the bounds on R m (3), but we may obtain some corollaries on edge Folkman numbers easily. Let us discuss only an example now. Suppose that there is a Ramsey coloring G of K n for r k (K3 ; K m ), where n = r k (K3 ; K m ) − 1. If s > k + 1 and n ≥ R s (3), then the edge Folkman number f(s −

6.4 Monotony conjecture on (k1 , k2 , . . . , k t ) graphs | 65

k, 3, m) ≤ R s (3) ≤ r k (K3 ; K m ) − 1. Otherwise, r k (K3 ; K m ) − 1 < R s (3). Therefore, if f(s − k, 3, m) > r k (K3 ; K m ) − 1, then r k (K3 ; K m ) − 1 < R s (3).

6.4 Monotony conjecture on (k1 , k2 , . . . , k t ) graphs We may call the following Conjecture 6.4.1 as Monotony conjecture. We have to say that this conjecture is much different from the other conjectures discussed in this book. It seems that it SHOULD NOT be difficult, but we can do nothing after proposing it. This fact makes us feel a little boring. We will propose a similar conjecture on van der Waerden numbers later in Conjecture 11.4.3 in Chapter 11, which we know nothing about how to solve it too. On the other hand, different from some famous conjectures, for instance, the Goldbach conjecture, we believe that the following Conjecture 6.4.1 MUST hold rather than maybe hold, but as told above, we can prove nothing now. Conjecture 6.4.1. If k 1 , . . . , k t are integers and k 1 ≥ ⋅ ⋅ ⋅ ≥ k t ≥ 3, then for any positive integer n < R(k 1 , . . . , k t ), there is a (k 1 , . . . , k t )-graph G of order n, and a vertex v in V(G), such that d1 (v) ≥ ⋅ ⋅ ⋅ ≥ d t (v), where d i (v) is the number of edges adjacent to v in color i in graph G for any i ∈ {1, . . . , t}. It will be interesting even if Conjecture 6.4.1 is proved only for Ramsey graphs, where n = R(k1 , . . . , k t )− 1 and graph G is a (k 1 , . . . , k t )-Ramsey graph. Now let us consider a problem related to this conjecture for bi-color Ramsey graphs. Let G be a (k−1, k)-Ramsey graph. Therefore |V(G)| = R(k−1, k)−1. We conjecture that δ(G) ≥ 12 (R(k − 1, k) − 2) . On the other hand, we can see that δ(G) ≤ R(k − 1, k − 1) − 1. All these mean that we conjecture that R(k − 1, k − 1) ≥ 12 R(k − 1, k) . Because R(k, k) ≤ 2R(k − 1, k), if the inequality above holds, then R(k − 1, k − 1) ≥ 14 R(k, k) , which is out of reach for us now. Based on the known data we can see that for i ∈ {4, 5, 6} this inequality holds, and we take it as a conjecture for k ≥ 7. Conjecture 6.4.2. For any integer k ≥ 7, R(k − 1, k − 1) ≥ (1/4)R(k, k). Walker proved that R(k, k) ≤ 4R(k, k − 2) + 2 in [291]. We cannot prove an inequality similar to that in Conjecture 6.4.2 because we cannot prove R(k, k −2) ≤ R(k −1, k −1), which is a subcase of Conjecture 5.2.4.

66 | 6 Multicolor classical Ramsey numbers

6.5 The Mathon–Shearer construction In [194] Mathon studied the lower bounds for multicolor classical Ramsey numbers based on so-called cyclotomic Ramsey colorings, i.e., multicolor Paley graphs. In [257], Shearer studied the bi-color case independently, and pointed out the method can be used on the multicolor case similarly. An improvement to Mathon’s cyclotomic Ramsey colorings was given in [305], in which Mathon’s results were improved for all multicolor cases. For instance, the best known lower bound for R(7, 7, 7) was obtained this way. More general, a related problem on Ramsey numbers is, if there is some graph classes that are easy to define and can show good lower bound for classical Ramsey numbers, including the bi-color and the multicolor ones.

6.6 More remarks There are more results in some references on classical Ramsey numbers, for instance, [312]. It is interesting to study if we can find new model of random graphs, that can help us to obtain better lower bounds for multicolor diagonal classical Ramsey numbers. Note that the Erdős–Rényi model of random graphs cannot work well for this topic. In this section, we will do some remarks on constructive methods and computing in Ramsey theory respectively.

6.6.1 Remarks on constructive methods The probabilistic method proves the existence of an objection without constructing it. The famous work of Erdős on the lower bound for R(k, k) is one of the earliest important applications of the probabilistic method. In the past decades, the probabilistic method has found many important applications in the development of combinatorics. Only in Ramsey theory, we can find many objections that we can prove the existence by the probabilistic method, but cannot give explicit constructions. For instance, the known lower bound for R(k, k) obtained by the probabilistic method, is much better than the best known constructive one. If we regard those existence proofs based on the probabilistic method as smart, then we may regard constructive proofs as thorough understanding. Furthermore, constructive methods and results are indispensable in applications. Of course, it is still unknown if there is a limit for constructive results that we can never surpass.

6.6 More remarks

| 67

6.6.2 Remarks on computing At the end of this section, let us discuss a little on computing in Ramsey theory. Should the work on computing upper and lower bounds for small Ramsey numbers or similar problems be encouraged? This question leads to the following discussion. More often than not, the research on algorithms should be encouraged. Of course, shallow or repetitive research should not be among them. It is a pity that some works on computing the bounds for small Ramsey numbers are like this, and some mathematicians may believe that all of them are so. Let us cite what Arnold said. We may say the same for some interesting computing in Ramsey theory. In 2005, at the beginning of [18], Arnold wrote: From the deductive mathematics point of view most of these results are not theorems, being only descriptions of several millions of particular observations. However, I hope that they are even more important than the formal deductions from the formal axioms, providing new points of view on difficult problems where no other approaches are that efficient.

In combinatorics, in particular during the development of Ramsey theory, the main reason that computing has not made a greater contribution to the development of the theory is the divorce between the theoretical research and the computing – we know there are very few people who are good at both of them. Maybe most mathematicians are good at neither of them. Even worse, many theorists look down on the works of computing type, and many computing experts are often do not have the ability or courage to deal with helpful theoretical thinking or necessary reading patiently. Currently, computer techniques are developing very quickly and disdaining computation seems inappropriate. Yes, computing the upper and lower bounds for small Ramsey numbers does not make enough of a contribution to the development of the theory. However, is this because it doomed to be so, or because it is too difficult and the contribution can only be enough when some great theoretical improvements occur after a long period of development? It seems too early to answer this question. We believe that the latter is true with a large probability. It is not difficult to find the similarity between some theoretical combinatorial mathematicians’ looking down on the computing experts, and the so-called main stream mathematicians’ looking down on combinatorial mathematicians.

7 Generalized Ramsey numbers In this chapter, we discuss the generalized Ramsey number. It is a generalization of the classical Ramsey number, for what the monochromatic subgraphs considered may not be complete graphs. There are many new papers on generalized Ramsey numbers, and more are published every year, most among which seem not to be interesting for most mathematicians not working on them, because more often than not, one needs different methods to match the related graphs for different generalized Ramsey numbers. It seems that most results on generalized Ramsey numbers do not have much influence on Ramsey theory and other parts of combinatorics. Therefore, in this chapter, we list only a few conjectures and problems on generalized Ramsey numbers. Note that among the new papers in Ramsey theory published in recent years, only a few are on classical Ramsey numbers, and there are not as many new papers on generalized Ramsey numbers as before. Of course, if readers were to tell us that for almost all topics in mathematics and for most mathematicians who do not work on them, these topics do not seem interesting, then we would have to say that they are (almost) right. Sometimes problems do not only occur on the side of topics, but also the group of mathematicians of our time. At least some seemingly not interesting problems may become interesting if we work hard and think deeply about them. However, this does not happen as often as it should. Turán numbers for different graphs are often used in studying generalized Ramsey numbers. As we know, both from a theoretical and a computational point of view, the problem of Turán numbers may be very difficult in many cases. In Section 7.1 we will discuss Frank Harary and generalized Ramsey numbers. We hope that the content in this section will not only familiarize more readers with the origin of generalized Ramsey numbers, but also make them consider similar problems among a larger range.

7.1 Frank Harary and generalized Ramsey numbers In an article titled “The future of graph theory”, Béla Bollobás wrote that we (mathematicians who research graph theory) must agree with Dieudonné that we do publish embryonic solutions of “problems without issue.” We believe that this often happens for some difficult problems in some papers on generalized Ramsey numbers. We have not found any negative remarks on generalized Ramsey numbers in any references. Let us cite what Frank Harary wrote in “A survey of generalized Ramsey theory” and “Adventures with Ramsey theory” on his discovery of generalized Ramsey numbers. Harary stated that his personal discovery of generalized Ramsey theory for graphs took place in October 1962. After hearing Erdős speak on R(m, n), Harary told https://doi.org/10.1515/9783110576702-007

70 | 7 Generalized Ramsey numbers

Erdős that “you can define R(F, H) for any two graphs F and H with no isolates, not necessary complete graphs.” Erdős replied that it would not be interesting. Harary said “It will” and Erdős said “It won’t!” Again, Harary said “It will” and Erdős insisted “It won’t!” So Harary did not work on it then. Eight years later, Harary shared this “generalized Ramsey theory” with Chvátal, and it led to four joint papers. Only later did they learn that other authors were independently discovering generalized Ramsey numbers for graphs, among them Burr, Cockayne, Gerencsér and Gyárfás, and Parsons. The paper of Gerencsér and Gyárfás, [132], published in 1967, is the earliest one among these works. Of course, the story of Harary is true, because Erdős knew it. It is obvious that Harary himself did not regard generalized Ramsey numbers as very interesting in October 1962, as he told Erdős, or he was not sure if it was important. In 1962, it was obvious for Harary that it was not difficult for him to write a paper (the first one) on generalized Ramsey numbers. Of course, he believed that the generalized Ramsey number is natural and that it is possible for other mathematicians to propose it independently. We may conjecture that he regarded the priority as unimportant because the generalized Ramsey number is not of much importance. The remark of Erdős that the generalized Ramsey number will not be interesting, is reasonable not only at that time but also later, even today. I think, if Wyle remarked on the generalized Ramsey theory today, he might ridicule with the tone of Gorden (who remarked on invariant theory), “Oh, this is surely very useful, on it people can write many papers.” Of course, we know that Erdős believed that generalized Ramsey numbers are interesting later. He wrote many papers on generalized Ramsey numbers with his co-authors, much more than the papers he wrote on classical Ramsey numbers. Erdős also proposed many problems and conjectures on generalized Ramsey numbers. Some people may believe that this is because most related easy works on classical Ramsey numbers have been done. This kind of idea has some truth in it. Therefore, more and more people will work on those problems that may be not so interesting, dreaming that new progress based on new idea will occur. However, on the other hand, we often find that some mathematicians make interesting progress based on some new, but simple ideas.

7.2 A generalization of a known result on classical Ramsey numbers The following theorem and its generalization were proved in [312]. Theorem 7.2.1. For k, l ≥ 3, let G be a (k, l; 2n)-graph, and suppose that for some partition V(G) = V1 ⋃ V2 the induced subgraphs G[V1 ] and G[V2 ] are isomorphic. Then, given any (s, 3; m)-graph, we have R(s, k, l) ≥ mn + 1 .

7.3 Some problems relative to R(C m , K n ) | 71

In most cases, it seems difficult to prove similar results on general Ramsey numbers. We can prove the following theorem. Theorem 7.2.2. Suppose that integers k ≥ 4 and l ≥ 3. Let G be a (K k − e, K l ; 2n)graph. Suppose that for some partition V(G) = V1 ⋃ V2 , the induced subgraphs G[V1 ] and G[V2 ] are isomorphic and K k−2 -free. Then for any integer s ≥ 3, we have R(K s , K k − e, K l ) ≥ (R(3, s) − 1)n + 1 . Similar to the other theorems like Theorem 7.2.1, we have the following result. If G is a (K k − e, K l ; 2n + 1)-graph and there is a vertex v ∈ V(G) and some partition V(G) − {v} = V1 ⋃ V2 , such that the induced subgraphs G[V1 ] and G[V2 ] are isomorphic and G[V1 ⋃{v}] is K k−2 -free, then we can obtain R(K s , K k − e, K l ) ≥ (R(3, s) − 1)n + s . All these results can be proved similarly to the proof of Theorem 7.2.1. For instance, we know there is a cyclic (K3 , K17 −e)-graph G of order 81. Therefore, for vertex 0 ∈ V(G), we have a partition V(G) − {v} = V1 ⋃ V2 , where V1 = {1, . . . , 40} and V2 = {41, . . . , 80}. Hence, the induced subgraphs G[V1 ] and G[V2 ] are isomorphic because G is a cyclic graph. By computing we know that cl(G[V1 ]) = 12. Therefore, we have R(K s , K17 − e, K3 ) ≥ 40(R(3, s) − 1) + s. This is only an example in which the lower bound seems weak. In fact, we know that R(K s , K17 − e, K3 ) ≥ R(3, s, 16) ≥ 4R(15, s) − 3, and by computing we know that it is larger than 40(R(3, s) − 1) + s. Among those inequalities on the relation between different Ramsey numbers that we can prove, most are weak. However, it seems not easy to improve them. It is not difficult to generalize Theorem 7.2.2 to the following case. Theorem 7.2.3. Suppose that integers k ≥ 4 and l ≥ 3. Let G be a (K k − e, K l 1 , . . . , K l t ; 2n)-graph. Suppose that for some partition V(G) = V1 ⋃ V2 , the induced subgraphs G[V1 ] and G[V2 ] are isomorphic and K k−2 -free. Then for any integer s i ≥ 3, i ∈ {1, . . . , m}, we have R(K s1 , . . . , K s m , K k − e, K l 1 , . . . , K l t ) ≥ (R(3, s1 , . . . , s m ) − 1)n + 1 . We can also generalize the result below Theorem 7.2.2 similarly to the case in which |V(G)| is odd. For some graphs, we may generalize Theorem 7.2.3 to more general cases. We will not discuss the details here.

7.3 Some problems relative to R(C m , K n ) The following conjecture proposed by Erdős is one of the most famous conjectures on generalized Ramsey numbers. Conjecture 7.3.1. R(C4 , K n ) = o(n2−ϵ ) for some ϵ > 0.

72 | 7 Generalized Ramsey numbers

There exist positive constants c1 and c2 such that 3

c1 (m 2 / log m) ≤ R(C4 , K n ) ≤ c2 (m/ log m)2 . The lower bound was obtained by Bohman and Keevash in [39], and the upper bound was reported in [56], where, in turn, credit was given to an unpublished work by Szemerédi from 1980. The following conjecture was proposed in [100, 118]. Conjecture 7.3.2. R(C m , K n ) = (m − 1)(n − 1) + 1 for m ≥ n ≥ 3, except n = m = 3. Although there are many partial results on Conjecture 7.3.2, it seems far from solved now. It seems that to prove Conjecture 7.3.2 completely, we need more general methods that work in all cases. The proved parts of this conjecture was surveyed in [229], including for n ≥ 4m + 2, m ≥ 3 [211] and other cases in which m is among {3, 4, 5, 6, 7}. In [14] it was shown that for every fixed integer k ≥ 3, there are two constants c1 , c2 such that m2 m2 c1 ≤ r k (C4 ; K m ) ≤ c2 , 2 (log m) (log m)2 where r k (H; K) denotes the generalized Ramsey number R(H1 , . . . , H k , K), and H i = H for all i ∈ {1, . . . , k}. This result should not be regarded as the end of this problem, because the constants c1 , c2 should be studied more, if possible. We may make the same remark for many similar problems and results. Note that r k (C4 ; K m ) ≤ c2 m2 /((log m)2 ) implies that R(C4 , K m ) ≤ c2 m2 /((log m)2 ), which matches the best known upper bound for R(C4 , K m ). In the other part of this section, let us discuss some problems relative to R(C4 , C4 , K4 , K4 ). More often than not, graphs and colorings obtained in studying the lower bounds for classical Ramsey numbers are not useful for studying upper bounds for vertex and edge Folkman numbers. At least, we cannot use them to obtain interesting results on Folkman numbers easily. Then it is natural to consider those graphs obtained by studying the lower bounds for general Ramsey numbers, to see if they work. Now let us discuss R(C4 , C4 , K4 , K4 ). It is only an example, and we may consider more Ramsey numbers similarly. We know that 87 ≤ R(C4 , C4 , K4 , K4 ) ≤ 179. The gap between the upper and lower bounds is large. For the upper bound, we cannot do better because we do not know enough about the Turán number ex(n, C4 ) for large n. The bounds for R(C4 , C4 , K m ) were studied in [14]. We may try to obtain a lower bound 88 for some Ramsey number R(C4 , C4 , K m ), where we hope m is small. Let G be a (C4 , C4 , K m )-coloring of K87 . Let G i be the subgraph of G induced by the edges in color i, where i ∈ {1, 2, 3}. Then we try to check whether G3 → (4, 4)e holds. If G3 󴀀󴀂󴀠 (4, 4)e , then we have improved the known lower bound for R(C4 , C4 , K4 , K4 ). If G3 → (4, 4)e and we obtain an upper bound for edge Folkman number Fe (4, 4; m).

7.4 The connected Ramsey number | 73

If m is small, Fe (4, 4; m) ≤ 87 may be interesting. We know that 22 ≤ Fe (4, 4; 17) ≤ 25, which were obtained by Nenov and Kelov, respectively. For m < 17, it seems that no interesting upper bound on Fe (4, 4; m) is known. Testing edge-arrowing is often difficult, and we hope that we can obtain a new lower bound for R(C4 , C4 , K4 , K4 ) this way. If we can, we may try a larger lower bound for (C4 , C4 , K m )-colorings similarly.

7.4 The connected Ramsey number In [281], Sumner found that it seems plausible that if the restriction that each of G and G be a connected graph is placed on the graphs considered, then we may not need to require R(F, H) vertices in G in order to insure that F ⊂ G or H ⊂ G. This turns out to often be the case. In order to investigate this situation, Sumner defined the connected Ramsey numbers as follows [281]. A graph G is totally connected if both G and its complement are connected. The connected Ramsey number rc (G, H) is the smallest integer k ≥ 4 so that if G is a totally connected graph of order k, then either F ⊂ G or H ⊂ G. In [281], Sumner showed that if neither of F nor H contains a bridge, then rc (F, H) = R(F, H), the usual generalized Ramsey number of F and H. We can see that rc (K s , K t ) = R(s, t) for integers s ≥ t ≥ 3. Note that a bridge is an edge separating its ends. In [281] Sumner determined the value of rc (P n , P m ) and proposed the following problem. Problem 7.4.1. Determine the connected Ramsey numbers for pairs of graphs at least one of which contains a bridge. In particular, determine rc (P n , C m ). Sumner also suggested to investigate the more general k-connected Ramsey numbers, which can be defined similarly for pairs of graphs [281]. Let the related k-connected Ramsey number be rc (F, H; k). Then, rc (F, H; 1) = rc (F, H), and rc (F, H; k) ≥ rc (F, H; k + 1). It may be interesting to consider rc (K s , K t ; k), where k ≥ s. If rc (K s , K t ; k) = R(s, t), and rc (K s , K t ; k + 1) < R(s, t), then there is a k-connected (s, t)-Ramsey graph and not a k + 1-connected (s, t)-Ramsey graph. Note that we need not consider rc (K s , K t ; k) for k ≤ s − 1 because we know that every (s, t)Ramsey graph is (s−1)-connected, and in such a case the related k-connected Ramsey number equals R(s, t). Suppose that G is K n with edges colored with one color among {color 1, . . . , color r}; G is r-totally connected if the subgraph induced by all edges in color i is connected for any i ∈ [r]. Therefore, 1-totally connected is same as totally connected in essence. We can generalize the connected Ramsey number to multicolor Ramsey numbers.

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7.5 The Ramsey–Turán problem In combinatorics, some problems and results are Ramsey-type and some are Turántype. For instance, some known results on the upper bounds for vertex Folkman numbers and van der Waerden numbers are Turán-type. We will discuss them in Chapters 8 and Chapter 11, respectively. We may also regard them as being of density type in these cases.

7.5.1 The Turán problem with bounded chromatic numbers When we research and compute the bounds for Folkman numbers, in particular for small Folkman numbers, sometimes we need to consider the Turán-type problem with constraint conditions on chromatic numbers and independence numbers. We propose the following general problem on the clique number, chromatic number, and degree as follows. Problem 7.5.1. Suppose that t and r are integers and t > r ≥ 3. If G is any K r -free graph of order n with chromatic number no smaller than t, study the minimum degree of G. For related problems in the general case, the lower bound for the chromatic number is a local restriction. This is because if the chromatic number of a subgraph is large enough, the other part of the graph is no longer restricted by the lower bound on the chromatic number, but the bounded upper bound for the clique number is global, and this leads us back to the problems of Ramsey–Turán type, in which some Ramsey type problems appear. More often than not, in computing we need the minimum degree rather than lower bound on the edge number. In [15], it was proved that if G is a K r -free graph of order n with chromatic number r, then δ(G) ≤ (n(3r − 7))/(3r − 4). For instance, if G is a K4 -free graph of order 20 with chromatic number 4, then δ(G) ≤ 12. However, for the K r -free graph with a larger chromatic number, the problem will be more difficult. We can find few references on this case. We need to consider similar problems in studying the lower bounds for small vertex Folkman numbers and edge Folkman numbers in Chapter 8, including Fe (3, 3; 4).

7.5.2 The Ramsey–Turán number Let H be a fixed forbidden graph and let f be a function of n. Denote by RT(n, H, f(n)) the maximum number of edges a graph G on n vertices can have without containing H as a subgraph and also without having at least f(n) independent vertices. In other words, RT(n, H, f(n)) is the maximum number of edges a (H, K f(n) )-graph G on n ver-

7.5 The Ramsey–Turán problem | 75

tices can have. The problem of estimating RT(n, H, f(n)) is one of the central questions of so-called Ramsey–Turán theory. Simonovits and Sós gave an excellent survey of Ramsey–Turán theory and mentioned some old and new interesting open questions [266]. In [279] Sudakov obtained new bounds for some Ramsey–Turán-type problems. These results give partial answers to some of the questions in [266]. As Sudakov pointed out, finding the Ramsey–Turán number of K2,2,2 remains an intriguing open problem, and even the following simpler question is unsolved. Problem 7.5.2. Decide if RT(n, K2,2,2 , o(n)) = o(n2 ) or not. We would like to propose another problem on the Ramsey–Turán number as follows. Problem 7.5.3. Study the exact value and bounds for RT(r2 + 1, K r+1 , r + 1). It is interesting to know whether RT(r2 + 1, K r+1 , r + 1) < ((r2 + 1)(r − 1)r)/2 holds.

8 Folkman numbers For k > max{a1 , . . . , a r }, let Fv (a1 , . . . , a r ; k) = {G | G → (a1 , . . . , a r )v and K k ⊈ G} , and Fv (a1 , . . . , a r ; k; n) = {G | G → (a1 , . . . , a r )v , |V(G)| = n, K k ⊈ G} . Graphs in Fv (a1 , . . . , a r ; k) are called (a1 , . . . , a r ; k)v graphs, and an (a1 , . . . , a r ; k)v graph of order n is called an (a1 , . . . , a r ; k; n)v graph. The vertex Folkman number is defined as Fv (a1 , . . . , a r ; k) = min{|V(G)| | G ∈ Fv (a1 , . . . , a r ; k)} . Suppose that cl(H i ) < k for any i ∈ {1, . . . , r}. The edge Folkman number Fe (a1 , . . . , a r ; k) and the general vertex Folkman number Fv (H1 , . . . , H r ; K k ) can be defined similarly to Fv (a1 , . . . , a r ; k). The edge Folkman number Fe (a1 , . . . , a r ; k) is a generalization of the classical Ramsey number R(a1 , . . . , a r ). In fact, it is clear that Fe (a1 , . . . , a r ; k) = R(a1 , . . . , a r ) , when k > R(a1 , . . . , a r ). If cl(H i ) < k for any i ∈ [r], then we can define the general edge Folkman number Fe (H1 , . . . , H r ; K k ) by generalizing Fe (a1 , . . . , a r ; k) to general graphs. More generally, we may define Fv (H1 , . . . , H r ; H) and Fe (H1 , . . . , H r ; H) similarly, where H is not isomorphic to any subgraph of H i for any i ∈ [r]. Note that Fe (H1 , . . . , H r ; H) does not exist in some cases. For instance, we know that Fe (K3 , K3 ; K4 − e) does not exist.

8.1 Some known results on Folkman numbers In 1967, Erdős and Hajnal [103] posed a problem asking for a construction of a K6 -free graph for which every coloring of the edges with two colors contains a monochromatic triangle. They pointed out that the question was raised by Galvin. Erdős and Hajnal also expected (but did not prove) that for every number of colors r there is a K4 -free graph for which every coloring of the edges with r colors contains a monochromatic triangle. In 1970, Folkman [121] proved that for positive integers k and a1 , . . . , a r , Fv (a1 , . . . , a r ; k) (Fe (a1 , a2 ; k)) exists if and only if k > max{a1 , . . . , a r } (k > max{a1 , a2 }). For edge Folkman numbers, Folkman’s method only works for two colors. The existence of Fe (a1 , . . . , a r ; k) was proved by Nešetřil and Rödl in [207] (also see [147]). If s1 = ⋅ ⋅ ⋅ = s r = s, then Fv (s1 , . . . , s r ; s + 1) is written as F(r, s, s + 1), and Fe (s1 , . . . , s r ; s + 1) is written as f(r, s, s + 1), and Fv (s1 , . . . , s r ; s + 1) is written as F(r, s, s + 1), and Fe (s1 , . . . , s r ; s + 1) is written as Fe (r, s, s + 1). https://doi.org/10.1515/9783110576702-008

78 | 8 Folkman numbers In [243], Rödl, Ruciński, and Schacht obtained a new upper bound on f(r, k, k +1), which is the best known upper bound for f(r, k, k + 1). We cite it in the following theorem. Theorem 8.1.1. For all integers r ≥ 2 and k ≥ 3, f(r, k, k + 1) ≤ 2c(k

4

log k+k3 r log r)

for some c independent of r and k. This theorem was proved by the probabilistic method by considering the random graph G(n, p) for p = Cn−2/(k+1) , where n = n(k, r) and C = C(n, k, r). In an old proof on the existence of f(r, k, k + 1), a graph G ∈ Fe (r, k, k + 1) was constructed, of which the chromatic number equals R r (k) (see [207], also [147]). However, in the proof of Theorem 8.1.1 in [243], the chromatic numbers of those related graphs in Fe (r, k, k + 1) may be much larger. It is not difficult to find some references on the chromatic numbers of random graphs. We can find the following theorem on the chromatic number of the random graph G(n, p) in [42]. Theorem 8.1.2. Let 0 < p < 1 be fixed, and set q = 1 − p and d = 1/q. Then a.e. G(n, p) is such that n log log n n 3 log log n (1 + ) ≤ χ(G(n, p)) ≤ (1 + ) . 2 logd n log n 2 logd n log n In particular, χ(G(n, p)) = (1 + o(1))n/(2 log d n) for a.e. G(n, p) We know that if G ∈ Fe (r, k, k + 1), then χ(G) ≥ R r (k). It is interesting to know whether there is a graph G ∈ Fe (r, k, k + 1) of order f(r, k, k + 1), such that χ(G) = R r (k). Similarly, we may also ask whether there is a graph H ∈ F(r, k, k+1) of order F(r, k, k+ 1), such that χ(H) = r(k −1)+1. That there is a graph H ∈ F(r, k, k +1) such that χ(H) = r(k − 1) + 1, was proved in [302] recently. The order of the related graph constructed in [302] is much larger than F(r, k, k + 1). It was proved by Burr [53] that it is NP-complete to test whether a given graph does not arrow (G, H) e for any fixed graphs G and H that are either 3-connected or isomorphic to K3 . Some problems on the complexity of generalized graph colorings, including the vertex Folkman number as a special case, have been studied by some researchers. For instance, Achlioptas [3] proved that if |V(G)| > 2, then G-free 2-colorability is NP-complete. There are much fewer references on Folkman numbers than on Ramsey numbers. Let us give an example and explain why. Let Q(n, c) be the minimum clique number over graphs with n vertices and chromatic number c. In [34], Q(n, c) was studied, with no references on Folkman numbers cited. It is not difficult to see that what was studied in [34] is equivalent to a special class of vertex Folkman numbers. In fact, we can see that if F(c − 1, 2, t + 1) ≤ n < F(c − 1, 2, t) , then Q(n, c) = t.

8.2 Small Folkman numbers

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Some people may believe that Folkman numbers are not as basic and interesting as Ramsey numbers. Either belonging to theoretic research or computing, there are two kinds of problems on Folkman numbers. One kind includes difficult problems, the other kind includes those problems believed to be not important, which may be either easy or difficult. If we consider only the computation of small cases, we can see that there are also much fewer references on Folkman numbers than on Ramsey numbers, partly because in most cases both lower and upper bounds for Folkman numbers are difficult to compute. For instance, if G is a given K9 -free graph and G → (8, 8)v , it may be not easy to prove this by computing. Only a few mathematicians and computer scientists study the computation on small Folkman numbers. Such a situation makes few mathematicians and computer scientists consider how to compute bounds for small generalized Folkman numbers, because they believe that other people, including both editors of mathematical journals and readers, will not regard this work as interesting. Note that there are many papers on generalized Ramsey numbers, among which some focus on computing and many seem not interesting. On the other hand, to obtain new lower bounds for some Ramsey numbers may be more feasible than for edge Folkman numbers. The upper bounds for some small vertex Folkman numbers are not difficult, but related results may be more difficult to publish in a good journal than similar results on Ramsey numbers. We believe that works on Folkman numbers will loom large in the future.

8.2 Small Folkman numbers In this section, we consider bounds and exact values of some small vertex and edge Folkman numbers. Note that Fv (k, k; k + 1) may be discussed in this section if k is small, no matter whether or not the exact value and known bounds on Fv (k, k; k + 1) are small. The best known upper bound on F(r, s, s + 1) is Cs2 log 4 s proved in [89]. We will discuss it more later. The best known lower bound on Fv (k, k; k + 1) is 4k − 1, which was proved in [307]. We can see that the difference between the known upper and lower bounds on Fv (k, k; k + 1) for some small k’s is large, and the case for small Fe (k, k; k + 1) is similar.

8.2.1 On F e (3, 3; 4) Let us now discuss Fe (3, 3; 4). It is one of the most famous small Folkman numbers. The upper bound on Fe (3, 3; 4) obtained by Folkman [121] is very large. Spencer [274], Linyuan Lu [191], and Dudek and Rödl [86], obtained better upper bounds. Among these works, the upper bound obtained by Dudek and Rödl is 941. This upper bound was proved based on a cyclic K4 -free graph G of order 941 defined based on 5th

80 | 8 Folkman numbers

residues of 941. By the way, it seems that α(G) is not large. Maybe we can obtain a good lower bound for some R(4, k) if we can compute α(G). Although G is a generalized Paley graph and has good properties, α(G) is difficult to compute because the order 941 is too large. Let us consider a conjecture on Fe (3, 3; 4) proposed by Geoffrey Exoo. Conjecture 8.2.1. G127 → (3, 3)e . Here G127 is the cubic Paley graph defined based on the cubic residues of 127. If we can prove that Conjecture 8.2.1 holds, then the upper bound for Fe (3, 3; 4) is improved to 127. We know that G127 is a (4, 12)-graph. Exoo suggested that even a 94-vertex induced subgraph of G127 , obtained by removing from it three disjoint independent sets of order 11, may still work. If this is true, then it implies that Fe (3, 3; 4) ≤ 94. We know that 19 ≤ Fe (3, 3; 4) ≤ 786, which was proved by Radziszowski and Xiaodong Xu in [233] and by Lange, Radziszowski and Xiaodong Xu in [177], respectively. The method used in [177] is not enough to tell whether G127 → (3, 3)e . In 2016, the lower bound 19 was improved to 20 by Bikov and Nenov [32]. As remarked in [32], the method in [233] can be used to prove Fe (3, 3; 4) ≥ 20, but the computation is much more than in [32].

8.2.2 A small example It is not difficult to turn the problem related to Conjecture 8.2.1 on G127 or other similar problems into a problem of proving a given system of equalities to be no solution, or similarly turn it into a problem of finding the minimum of a polynomial in several variables. However, both of these seem be no help in solving the problem. This is similar to what Lovasź discussed in his interesting article [190] titled “Discrete and continuous: Two sides of the same?” We can take C5 → (K2 , K2 )v as an example to show how to prove an upper bound for a Folkman number through an algebraic method. We know that C5 → (K2 , K2 )v . We will not consider it by the easiest method here. Suppose that V(C5 ) = ℤ5 . Let us give C5 a 2-vertex coloring, that is, let f map v i to either 0 or 1 in ℤ2 , where i ∈ ℤ5 . Let f i = f(v i ). Because we want a red-blue coloring of V(C5 ) without monochromatic K2 , we may suppose that f i ≠ f i+1 holds for any i ∈ ℤ5 . Hence, f i + f i+1 = 1 and f i2 = f i for any i ∈ ℤ5 . Therefore, f i f i+1 = 0 for any i ∈ ℤ5 . Note that all the following computations are done in ℤ2 . We have Π i∈ℤ5 (f i + f i+1 ) = 1. We can see that (f1 + f2 )(f2 + f3 ) = f1 f2 + f1 f3 + f22 + f2 f3 = f1 f3 + f2 . Similarly, (f3 + f4 )(f4 + f5 ) = f3 f5 + f4 . Therefore, (f1 + f2 )(f2 + f3 )(f3 + f4 )(f4 + f5 ) = (f1 f3 + f2 )(f3 f5 + f4 ) = f2 f4 . Thus, Π i∈ℤ5 (f i + f i+1 ) = f2 f4 (f5 + f1 ) = 0. It is a contradiction. Hence, C5 → (K2 , K2 )v .

8.2 Small Folkman numbers |

81

We may also prove it another way. Let h i = f i + f i+1 − 1. Suppose that A = h2 + h4 + h5 − h1 − h3 and B = h1 + h3 + h4 − h2 − h5 . Then, we have A = 2f5 − 1 and B = 2f4 − 1. Therefore, AB − 4f4 f5 + 2(f4 + f5 ) = 1. Hence, AB − 4f4 f5 + 2h4 = −1 because AB = h24 − (h2 + h5 − h1 − h3 )2 , (h2 + h5 − h1 − h3 )2 − h24 + 4f4 f5 − 2h4 = 1 . This is a proof of the non-solution of the equation system {h i = 0 (i ∈ ℤ5 ) { f f =0, {4 5 given by Hilbert’s Nullstellensatz. By this proof we can see that C5 → (K2 , K2 )v . It is interesting to know what can we obtain if we deal with G127 → (K3 , K3 )e similarly. It is far from our reach now. We suggest that the interested reader consider how to prove K6 → (3, 3)e similarly as an exercise, which is much easier. Now let us consider more small Folkman numbers. We may do some computation on Problem 8.2.1, Problem 8.2.2, Problem 8.2.3, and Problem 8.3.2.

8.2.3 Bounds on Fe (K4 − e, K4 − e; K4 ) Some interesting problems were suggested in [230]. Let us cite the following problem from among them. Note that K4 − e is a graph obtained by deleting an edge from K4 . Problem 8.2.1. Compute the value or bounds for Fe (K4 − e, K4 − e; K4 ). We know that 20 ≤ Fe (3, 3; 4) ≤ Fe (K4 − e, K4 − e; K4 ) ≤ 30,193 . The upper bound was observed by Linyaun Lu in [191]. We may try to give Fe (K4 −e, K4 − e; K4 ) a new lower bound better than 20, before doing similar work on Fe (3, 3; 4). Note that Fe (K3 , K4 − e; K4 ) ≥ Fv (K4 − e, K4 − e; K4 ), and the value of Fv (K4 − e, K4 − e; K4 ) has not been determined. Furthermore, we can prove Fe (K3 , K4 − e; K4 ) ≥ Fv (K4 − e, K4 − e; K4 ) + 1 similarly to Fe (3, k; k + 1) ≥ Fv (k, k; k + 1) + 1, which can be generalized to more general cases without difficulty. It is interesting to know if Fe (3, 3; 4) < Fe (K4 − e, K4 − e; K4 ) without computing. More generally, we propose the following problem. Problem 8.2.2. Does Fe (k − 1, k − 1; k) < Fe (K k − e, K k − e; K k ) hold for every integer k ≥ 4? Furthermore, we may consider if Fe (k − 1, k − 1; k) < Fe (K k−1 , K k − e; K k ) and Fe (K k−1 , K k − e; K k ) < Fe (K k − e, K k − e; K k ) hold.

82 | 8 Folkman numbers 8.2.4 Bounds on Fv (4, 4; 5) For Folkman numbers of the form Fv (k, k; k + 1), we know that Fv (2, 2; 3) = 5, Fv (3, 3; 4) = 14 (see [220]) and 17 ≤ Fv (4, 4; 5) ≤ 23 (see [303]). We also know that Fv (2, 2, 2, 4; 5) ≥ 17. In [172], N. Kolev proved the following multiplicative inequality for vertex Folkman numbers, which was proved independently in [304]. In [304], more constructive results on the upper bound for Fv (k, k; k + 1) were proved, which improved earlier known bounds but is much weaker than the best known upper bound for Fv (k, k; k+1) in [85]. Theorem 8.2.1. If max{a1 , . . . , a r } ≤ a and max{b 1 , . . . , b r } ≤ b, then Fv (a1 b 1 , . . . , a r b r ; ab + 1) ≤ Fv (a1 , . . . , a r ; a + 1)Fv (b 1 , . . . , b r ; b + 1) . We can see that Fv (6, 6; 7) ≤ 70 and Fv (8, 8; 9) ≤ 115, because by Theorem 8.2.1 we have Fv (6, 6; 7) ≤ Fv (3, 3; 4)Fv (2, 2; 3) = 14 × 5 = 70 , and Fv (8, 8; 9) ≤ Fv (4, 4; 5)Fv (2, 2; 3) ≤ 23 × 5 ≤ 115 . It seems that we are far from determining the exact value of Fv (k, k; k + 1) for k ≥ 5. We propose the following problem as an interesting small case. Problem 8.2.3. Compute the value or bounds for Fv (4, 4; 5). We know that G13 , the generalized Paley graph of order 13 defined based on the cubic residues of 13, is the unique (3, 5)-Ramsey graph, the complement of which is the unique graph in Fv (3, 4; 5; 13) (see [303]). This graph may be used in improving the known lower bound for Fv (4, 4; 5). It does not seem easy to know whether there is a graph G ∈ Fv (4, 4; 5), in which there is an independent set such that the induced subgraph obtained by deleting this independent set is isomorphic to G13 . In particular, it is interesting to know whether there is such a K5 -free graph G of order 22. If there is, then we can improve the known upper bound for Fv (4, 4; 5) to 22. The general problem behind Problem 8.2.3 is: give a general method to obtain interesting lower and upper bounds for vertex Folkman numbers. This is difficult for us now even for small vertex Folkman numbers, let alone the similar problem on edge Folkman numbers, which seems much more difficult.

8.2.5 More problems on small vertex Folkman numbers There are many unsolved topics on small vertex Folkman numbers. In most cases, both the lower and the upper bounds are not easy to compute. Let us list some of them in the following problem.

8.3 Bounds on some small generalized Folkman numbers

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Problem 8.2.4. Study the lower bounds for F(3, 3, 4), F(4, 3, 4), F(5, 3, 4). It is not difficult to prove the following theorem. Theorem 8.2.2. If b > a ≥ 2, then a F(a, k, k + 1) ≤ 1 + ⌊ (F(b, k, k + 1) − 1)⌋ . b Proof. For any graph G ∈ F(b, k, k + 1), suppose that |V(G)| = F(b, k, k + 1). Let G1 = G−v for some v ∈ V(G). Therefore G−v 󴀀󴀂󴀠 (b, k, k+1)v . Suppose that U1 , . . . , U b are b disjoint subsets of V(G) − v such that G[U i ] is K k -free for any i ∈ {1, . . . , b}, and ⋃ bi=1 U i = V(G) − v. Suppose that |U1 | ≤ ⋅ ⋅ ⋅ ≤ |U b |. It is not difficult to see that if ⋃ ai=1 U i ∪{v} = U, then G[U] → (a, k, k+1)v . Because G[U] is K k+1 -free and |V(G[U])| = |U| ≤ 1 + ⌊(a/b)(F(b, k, k + 1) − 1)⌋, we have F(a, k, k + 1) ≤ 1 + ⌊(a/b)(F(b, k, k + 1) − 1)⌋. Therefore, F(4, 3, 4) ≥ (4/3)(F(3, 3, 4) − 1), F(5, 3, 4) ≥ (5/3)(F(3, 3, 4) − 1) and F(5, 3, 4) ≥ (5/4)(F(4, 3, 4) − 1); F(3, 3, 4) = F v (3, 3, 3; 4) ≤ 66 was proved in [76]. By Fv (2, 3, 3; 4) ≥ 20 proved in [32], we can see that Fv (3, 3, 3; 4) ≥ 24, and if Fv (3, 3, 3; 4) = 24, then every graph G ∈ F(3, 3, 3; 4; 24) is a (4, 5)-Ramsey graph. Maybe we can obtain a better lower bound if we analyze the minimum degrees of related graphs in detail.

8.3 Bounds on some small generalized Folkman numbers In this section, let us consider some problems on the edge Folkman number of the form Fe (H1 , H2 ; K k+1 ), where cl(H i ) ≤ k for any i ∈ {1, 2}. Before we discuss more problems on generalized Folkman numbers, let us cite some interesting results of Yusheng Li and Qizhong Lin from [179]. In [179], for fixed m ≥ 3, it was shown that c(

(m+1)/2 n ≤ Fe (K m , K n,n ; K m+1 ) ≤ (m − 1)(n m−1 + n − 1) + 1 , ) log n

where c = c(m) is a positive constant. Here, the upper bound was proved by proving that K m (N, . . . , N, (m − 1)(n − 1) + 1) → (K m , K n,n )e , where K m (n1 , . . . , n m ) denotes the complete m-partite graph, in which the i-th part has n i vertices. In particular, there exists a positive constant c such that cn2 / log n ≤ Fe (K3 , K n,n ; K4 ) ≤ 2n2 + 2n + 1 , for sufficiently large n. Lin and Li proved R(K3 , K n,n ) ≥ cn2 / log n in [186] by extending a method of Bohman in [37].

84 | 8 Folkman numbers

In [179], it was also proved that Fe (K m , T n ; K m+1 ) ≤ m2 (n − 1) , for all n, m ≥ 2, where T n is any tree on n vertices. It is interesting to know the exact value of Fe (K m , T n ; K m+1 ).

8.3.1 On Fv (K3 − e, K3 − e; K3 ) We can prove Fv (K3 − e, K3 − e; K3 ) ≤ 9 without much difficulty with pencil and paper as in the following theorem. Theorem 8.3.1. Fv (K3 − e, K3 − e; K3 ) ≤ 9. Proof. Let G be a graph of order 9, and V(G) = {v i | i ∈ ℤ5 } ⋃{u i | i ∈ ℤ5 − {0}}. Let E1 = {v i v i+1 | i ∈ ℤ5 }, E2 = {u i v i+1 | i ∈ ℤ5 − {0}} ⋃{u i v i−1 | i ∈ ℤ5 − {0}}, and E3 = {u i u i+1 | i ∈ ℤ5 − {0, 4}}. Suppose that E(G) = E1 ⋃ E2 ⋃ E3 . We can see that G is K3 -free, because G is isomorphic to the subgraph of C5 [E2 ] obtained by deleting any vertex. It is not difficult to prove that G → (K3 − e, K3 − e; K3 )v . Therefore, we have Fv (K3 − e, K3 − e; K3 ) ≤ 9. In most problems in this book, we consider only generalized Ramsey numbers and generalized Folkman numbers for connected graphs. Now, let us consider the following cases of generalized vertex Folkman numbers, where non-connected graphs are included. Let us consider an example; Fv (E2 , H1 , H2 ; K k+1 ) denotes the smallest positive integer n such that there is a K k+1 -free graph G of order n, G − v → (H1 , H2 )v for any v ∈ V(G). Van der Waerden numbers will be discussed in Chapter 11 in this book; E2 in Fv (E2 , H1 , H2 ; K k+1 ) is a little similar to an arithmetic progression of two terms in the van der Waerden number W(2, a1 , . . . , a r ) studied by some people. It is not difficult to see that Fv (H1 , H2 ; K k+1 ) + 1 ≤ Fv (E2 , H1 , H2 ; K k+1 ) ≤ Fv (K2 , H1 , H2 ; K k+1 ) . In fact, it is not difficult to prove that Fv (E2 , P3 , P3 ; K3 ) ≤ 10, similarly to the proof of Theorem 8.3.1, which implies Theorem 8.3.1. Let us propose the following problem. Problem 8.3.1. Suppose that V(H1 ) ≥ V(H2 ) ≥ 3, and cl(H1 ) ≤ k, cl(H2 ) ≤ k. Does Fv (H1 , H2 ; K k+1 ) + 1 < Fv (E2 , H1 , H2 ; K k+1 ) always hold if both H1 and H2 are connected graphs?

8.3 Bounds on some small generalized Folkman numbers

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If Fv (H1 , H2 ; K k+1 )+1 = Fv (E2 , H1 , H2 ; K k+1 ), then there is a graph G ∈ Fv (E2 , H1 , H2 ; K k+1 ) such that |V(G)| = Fv (E2 , H1 , H2 ; K k+1 ) and G − v → (H1 , H2 )v for any v ∈ V(G). It is interesting to know whether the gap between Fv (H1 , H2 ; K k+1 ) and Fv (E2 , H1 , H2 ; K k+1 ) is always very small. We suggest obtaining an interesting upper bound for the gap between them first. Let us consider the lower bound for Fv (K3 − e, K3 − e; K3 ). We consider it not because it is important to know its exact value, but because we like to show an example for which it seems that we cannot find a simple proof for such a small result. Our proof need not be the easiest one. Let us consider the maximum degree of a graph of order 8 in Fv (P3 , P3 ; K3 ) now. Note that P3 = K3 − e. Theorem 8.3.2. If graph G → (P3 , P3 ; K3 )v and |V(G) = 8, then ∆(G) < 4. Proof. Suppose that graph G → (P3 , P3 ; K3 )v and the order of G is 8. It is not difficult to see ∆(G) ≤ 4, because otherwise for any vertex v ∈ V(G) with degree larger than 4, by V1 = N(v) and V2 = V(G) − V1 , we have a (P3 , P3 )v -coloring. Now let us prove that ∆(G) < 4. Suppose that d(v) = 4 for some vertex v ∈ V(G), and the neighbors of v are {v i | 1 ≤ i ≤ 4}. Then, we can see that the subgraph induced by three non-neighbors of v must be isomorphic to P3 . Suppose that the non-neighbors of v are {u i | 1 ≤ i ≤ 3} and u 1 u 2 , u 1 u 3 ∈ E(G). Suppose that u i0 and v have at most one common neighbor for some i0 ∈ {1, 2, 3}. Then, we can obtain a (P3 , P3 )-coloring by coloring v ∪({u i | 1 ≤ i ≤ 3}−{u i0 }) with red and other vertices with blue. Hence, every vertex among {u i | 1 ≤ i ≤ 3} has at least two common neighbors with v. Because G is K3 -free, u 1 and u i do not have common neighbors for any i ∈ {2, 3}. So, we may suppose that u 1 v1 , u 1 v2 , u 2 v3 , u 2 v4 , u 3 v3 , u 3 v4 ∈ E(G). If we color v, v1 , u 2 , u 3 with red and color other vertices with blue, then we can obtain a (P3 , P3 )-coloring and we can see that G 󴀀󴀂󴀠 (P3 , P3 )v . Therefore, ∆(G) < 4. It is not difficult to know there is not (3, 4)-Ramsey-graph that arrows (P3 , P3 ; K3 )v . In fact, we need only check if G → (P3 , P3 ; K3 )v , where G is the (3, 4)-Ramsey-graph of size 12. Note that any (3, 4)-Ramsey graph is a subgraph of graph G. It is not difficult to prove that Fv (K3 − e, K3 − e; K3 ) > 8 by Theorem 8.3.2 and more detailed computation. We omit the details here. Now we have the following Theorem 8.3.3. It is interesting to know whether there is a simple proof for Fv (K3 − e, K3 − e; K3 ) ≥ 9. Theorem 8.3.3. Fv (K3 − e, K3 − e; K3 ) = 9.

8.3.2 Bounds on Fv (K4 − e, K4 − e; K4 ) Let us consider the following problem on Fv (K4 − e, K4 − e; K4 ). Problem 8.3.2. Compute the value or bounds for Fv (K4 − e, K4 − e; K4 ).

86 | 8 Folkman numbers Bikov and Nenov proved that 20 ≤ Fv (2, 3, 3; 4) ≤ 24 in [32]. Maybe Problem 8.3.2 is not difficult. For instance, can the graph used to give upper bound 24 for Fv (2, 3, 3; 4) be used to improve the upper bound for Fv (K4 − e, K4 − e; K4 )? Maybe we should study the exact value of Fv (K3 , K4 −e; K4 ) before we study the lower bound for Fv (K4 −e, K4 − e; K4 ). It seems that the following Problem 8.3.3 is not solved. We can see that Fv (K4 − e, K4 − e; K4 ) ≥ Fv (K3 , K4 − e; K4 ) ≥ Fv (3, 3; 4) = 14 . However, we do not know if Fv (K4 − e, K4 − e; K4 ) > Fv (3, 3; 4) without computing. It is not difficult to test if any graph in Fv (3, 3; 4; 14) arrows (K4 − e, K4 − e; K4 )v or (K4 − e, K3 ; K4 )v . More generally, we propose the following problem. Problem 8.3.3. Does Fv (K k − e, K k − e; K k ) > Fv (k − 1, k − 1; k) hold for every integer k ≥ 4? It is interesting and difficult to give a small upper bound for the difference between Fv (K k − e, K k − e; K k ) and Fv (k − 1, k − 1; k). We propose the following problem here. Problem 8.3.4. Is there a positive constant c such that Fv (K k − e, K k − e; K k ) − Fv (k − 1, k − 1; k) ≤ ck ? We have no idea how to answer this problem. If there is a constant c1 such that Fv (k − 1, k − 1; k) ≤ c1 k, then by Fv (K k − e, K k − e; K k ) ≤ 3Fv (k − 1, k − 1; k) we have Fv (K k − e, K k − e; K k ) − Fv (k − 1, k − 1; k) ≤ 2Fv (k − 1, k − 1; k) ≤ 2c1 k , and the answer to Problem 8.3.4 should be positive. Of course, we do not know if Fv (k− 1, k − 1; k) has a linear upper bound.

8.4 On the generalized Ramsey numbers of Erdős and Rogers Suppose f s,t (n) = min{max{|S| : S ∈ V(H) and H[S] contains no K s }}, where the minimum is taken over all K t -free graphs H of order n (see [88]). In [89], it was proved that for every integer s ≥ 2, there is a positive constant c = c(s) so that for every integer n, f s,s+1(n) ≤ cn2/3 . Based on this result, Dudek and Rödl [89] proved the following theorem, when the asymptotic is taken in r. Theorem 8.4.1. For every positive integer s, there is a positive constant c = c(s) such that for every integer r, F(r, s, s + 1) ≤ cr3 .

8.4 On the generalized Ramsey numbers of Erdős and Rogers

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87

More importantly, the following theorem on the upper bound for F(r, s, s + 1) was also proved in [89]. Theorem 8.4.2. For a given positive integer r there exists a constant C = C(r) such that for every s the vertex Folkman number satisfies F(r, s, s + 1) ≤ Cs2 log 4 s. In [87] the following problem was proposed. Because it is difficult to obtain good lower bounds for Folkman numbers, only little known data exists. Problem 8.4.1. Decide if the ratio (F(r, s, s + 1))/s tends to infinity together with s. In [85] Dudek, Retter, and Rödl proved that for every s ≥ 3, there exists a constant c s such that 2 f s;s+1(n) ≤ c s (log n)4s n1/2 , which is best possible up to a polylogarithmic factor. Krivelevich proved that f s,s+1 (n) ≥ Ω(log log n)1/2 n1/2 (see [88]). The best known lower bound, observed by Dudek and Mubayi, stands at f s;s+1(n) ≥ c󸀠s (

n log n 1/2 . ) log log n

The most important problem in this section is the following. Problem 8.4.2. Decide f s;s+1(n) and Fv (s, s; s + 1). We know that the best known upper bound for F v (s, s; s + 1) was given by the smallest n such that f s;s+1(n) < n/2. It is interesting to know if the exact value of Fv (s, s; s+1) is much smaller than that obtained by f s;s+1 (n). We conjecture that it is, as in the following conjecture. In general, we would like to know whether the density methods such as the one based on f s;s+1 (n) can give us good bounds for vertex Folkman numbers, in particular, Fv (s, s; s + 1). Conjecture 8.4.1. Let h(s) be the smallest n such that f s;s+1(n) < n/2. There is a constant ϵ ∈ (0, 1) and positive integer N = N(ϵ) such that for any integer s > N, Fv (s, s; s + 1) ≤ (1 − ϵ)h(s) . On the other hand, it is interesting to consider the following problem. Problem 8.4.3. Study the limit of (F(r, k, k + 1))/r for given k when r → ∞. From Theorem 8.2.2 we can see that if b > a, then (F(b, k, k + 1) − 1)/b ≥ (F(a, k, k + 1) − 1)/a. Therefore, lim r→∞ (F(r, k, k + 1))/r = limr→∞ (F(r, k, k + 1) − 1)/r exists for any given k ≥ 2 and may equal ∞. It is not difficult to see that F(ar, k, k + 1) − 1 ≥ a(F(r, k, k + 1) − 1) for any positive integer a. It is not difficult to improve this inequality. In fact, we can prove the following theorem.

88 | 8 Folkman numbers Theorem 8.4.3. For any integers k, a, r ≥ 2, F(ar, k, k + 1) ≥ aF(r, k, k + 1). Let F(ar, k, k + 1) = n. For any G ∈ F(ar, k, k + 1) of order n and any v ∈ V(G), let H be the graph induced by all non-neighbors of v in G. Therefore, H ∈ F(ar − 1, k, k + 1). For any V1 ⊂ V(H) such that |V1 | = F(r, k, k + 1) − 1, G[{v} ∪ V1 ] 󴀀󴀂󴀠 (r, k, k + 1) and the subgraph of G induced by V(G)−({v}∪V1 ) arrows (ar−r, k, k+1)v , of which the order is n−F(r, k, k+1) ≥ F(ar−r, k, k+1). Hence F(ar, k, k+1)−F(r, k, k+1) ≥ F(ar−r, k, k+1). Similarly, we have that F(ar, k, k + 1) ≥ aF(r, k, k + 1). The following problem was proposed in [85]. Problem 8.4.4. For all t > s ≥ 3, is limn→∞ (f s+1,t+1(n))/(f s,t (n)) = infinite? As shown above, the problem of the bounding of vertex Folkman numbers has been studied by some authors. In [84], some related results were extended to uniform hypergraphs. We know that in [243], Rödl, Ruciński, and Schacht obtained the best known upper bound on the edge Folkman number f(r, k, k + 1). Now, let us cite the following definition and result on hypergraph Folkman numbers from [243]. Given three positive integers h, k, and r, the h-uniform Folkman number f h (k; r) is defined to be the minimum number of vertices in an h-uniform hypergraph H such (h) (h) (h) that H → (K k )r but H is K k+1 -free. Here, K k stands for the complete h-uniform hypergraph on k vertices, that is, one with C hk edges. The finiteness of hypergraph Folkman numbers was proved by Nešetřil and Rödl in [207] (Colloary 6, page 206). Rödl, Ruciński, and Schacht believe that their quantitative approach on edge Folkman numbers in [243] should also provide an upper bound on the hypergraph Folkman numbers f h (k; r), exponential in a polynomial of k and r. Let us propose the following problem on hypergraph Folkman numbers. Problem 8.4.5. Study the exact value and bounds for the hypergraph Folkman number f h (k; r).

8.5 Other problems on Folkman numbers Nenov [206] proposed a problem on the monotony of Fv (k, k; k + 1). Conjecture 8.5.1. Fv (k, k; k + 1) ≤ Fv (k + 1, k + 1; k + 2). Because Fv (2, 2; 3) = 5, Fv (3, 3; 4) = 14 and Fv (4, 4; 5) ≥ 17, we know that Conjecture 8.5.1 holds for k ∈ {2, 3}. Let us consider the case k = 4. We know that 17 ≤ Fv (4, 4; 5) ≤ 23; there is no published result on the lower bound for Fv (5, 5; 6). If there is a graph G ∈ Fv (5, 5; 6; 21), then by R(3, 6) = 18 we can find two disjoint independent sets of order 3, V1 and V2 , in V(G). Let G1 be G − (V1 ∪ V2 ). We can see that G1 → (3, 5)v because G → (5, 5)v . We can see that |V(G1 )| = 15, and by Fv (3, 5; 6) = 16 we know that G1 󴀀󴀂󴀠 (3, 5)v . This is a contradiction. Hence, Fv (5, 5; 6) ≥ 22.

8.5 Other problems on Folkman numbers

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Suppose that Fv (5, 5; 6) = 22, and G is any graph in Fv (5, 5; 6; 22). If V1 and V2 are two disjoint independent sets of order 3 in V(G), then G1 = G − (V1 ∪ V2 ) ∈ Fv (3, 5; 6; 16). However, we do not know if Fv (5, 5; 6) ≥ Fv (4, 4; 5). Maybe we can prove Conjecture 8.5.1 only when we can prove A(k) ≤ Fv (k, k; k + 1) ≤ B(k) , for some A(k) and B(k), such that B(k) ≤ A(k + 1). Now such a result is far from reach even if it is true. In fact, for general upper bounds on vertex Folkman numbers, we know very few results other than what was cited in the theorems in Section 8.4, in particular Theorem 8.4.2, which are not related to inequalities on different Folkman numbers. It does not seem easy to prove that lim (Fe (3, k; k + 1) − Fv (k, k; k + 1)) = ∞ .

k→∞

We know that Fe (3, k; k + 1) > Fv (k, k; k + 1) was proved in [307], and it is not difficult to improve this inequality by 1. Based on Fv (3, 3, 3; 4) ≤ 66, Fe (3, 3, 3; 8) ≤ 1 + Fv (6, 6, 6; 7) ≤ 727 was proved in [76]. Maybe this upper bound for Fe (3, 3, 3; 8) is much larger than the exact value. We can propose the following problem on classical Ramsey numbers here, which may be useful in studying the bounds for Folkman numbers. Problem 8.5.1. Is there is a positive integer k such that R(3, 3, 3, k) < R(8, k)? We do not know if there is a positive integer k0 such that R(3, 3, 3, k) < R(8, k) holds for any k ≥ k0 . Based on the known bounds on related Ramsey numbers, it is easy to know that if there is such an integer k, then k ≥ 5. We know that 101 ≤ R(5, 8) ≤ 216 and 162 ≤ R(3, 3, 3, 5) ≤ 286, where R(3, 3, 3, 5) ≤ 1 + 152 + 44 + 44 + 44 + 1 = 286. Here R(3, 3, 5) ≤ 45 and R(3, 3, 3, 4) ≤ 153 are used. If R(8, 5) > R(3, 3, 3, 5), then Fe (3, 3, 3; 8) ≤ R(3, 3, 3, 5) < R(8, 5) ≤ 216, because for any (8, 5)-graph H of order R(3, 3, 3, 5), H → (3, 3, 3; 8)e holds. Suppose that max{R(a − 1, b), R(a, b − 1)} < k − 1. If G is any graph in Fv (R(a − 1, b), R(a, b − 1); k − 1), then K1 + G → (a, b)e . Therefore, Fe (a, b; k) ≤ 1 + Fv (R(a − 1, b), R(a, b − 1); k − 1) . This is a well-known inequality, and it can be generalized to the multicolor case. We know that Fe (3, 3, 3; 8) ≤ 1 + Fv (6, 6, 6; 7). As we can see, the method used in studying the upper bound for Fe (3, 3, 3; 8) cannot be used in studying Fe (3, 3, 3; 7) directly. Let G = K3 + C5 . We know that Graham proved Fe (3, 3; 6) ≤ 8 based on G in [142]. It is a small exercise for the readers to tell whether this graph is the unique K6 -free graph that arrows (3, 3; 6)e . We can see that Fe (3, 3, 3; 7) ≤ 1 + Fv (G, G, G; K6 ). Our problem is as follows.

90 | 8 Folkman numbers Problem 8.5.2. Study the upper bounds for Fe (3, 3, 3; 7) and Fv (G, G, G; K6 ), where G = K3 + C5 . We do not know if Fe (3, 3, 3; 7) is much smaller than Fv (G, G, G; K6 ) + 1. All known methods seem to be insufficient to study this kind of problems. We can see that Fv (G, G, G; K6 ) ≥ Fv (5, 5, 5; 6) because K5 is a subgraph of graph G. Therefore, Fv (G, G, G; K6 ) cannot be small. It is interesting to know whether for any H ∈ Fe (3, 3; 6), Fv (H, H, H; K6 ) ≥ Fv (G, G, G; K6 ) holds, where G = K3 + C5 .

8.6 F v (3, k; k + 1) and other vertex Folkman numbers We know that (Fv (4, 2k; 2k + 1))/(Fv (2, k; k + 1)) ≤ 5. If k = 2, then (Fv (4, 2k; 2k + 1))/(Fv (2, k; k + 1)) = (Fv (4, 4; 5))/(Fv (2, 2; 3)) ≤ 23/5. If k = 3, then (Fv (4, 2k; 2k + 1))/(Fv (2, k; k + 1)) = (Fv (4, 6; 7))/(Fv (2, 3; 4)) = (Fv (4, 6; 7))/7. We know that Fv (4, 6; 7) ≤ 35, and we wish to obtain a better upper bound for it. Note that Fv (3, 6; 7) ≤ 18, and is much smaller than 35. For the general case, we propose the following conjecture. Conjecture 8.6.1. limk→∞ (Fv (4, 2k; 2k + 1))/(Fv (2, k; k + 1)) exists. Because Fv (2, k; k + 1) = 2k + 1, so what we conjecture above is that limk→∞ (Fv (4, 2k; 2k + 1))/k exists. It is not difficult to prove the following inequality on Fv (4, 2k; 2k + 1) constructively. Theorem 8.6.1. If k is an integer and k ≥ 2, then Fv (4, 2k; 2k + 1) ≤ 2Fv (3, k; k + 1) + 5k + 2 . Proof. Let H1 ∈ Fv (3, k; k + 1) and H2 ∈ Fv (2, k; k + 1), where |V(H1 )| = Fv (3, k; k + 1) and |V(H2 )| = Fv (2, k; k + 1) = 2k + 1. Let G0 = K k . Suppose that G1 and G2 are isomorphic to H1 , and G3 and G4 are isomorphic to H2 . Let V(G) = ⋃4i=0 V(G i ). Therefore, 4

|V(G)| = ∑ |V(G i )| = k + 2Fv (3, k; k + 1) + 2Fv (2, k; k + 1) . i=0

Hence |V(G)| = 2Fv (3, k; k + 1) + 5k + 2. Suppose that E1 = {uv | u ∈ V(G0 ), v ∈ V(G1 ) ∪ V(G2 )}, E2 = {uv | u ∈ V(G1 ), v ∈ V(G3 )}, E3 = {uv | u ∈ V(G2 ), v ∈ V(G4 )}, E4 = {uv | u ∈ V(G3 ), v ∈ V(G4 )}. Let E(G) = ⋃4i=0 E(G i ) ∪ ⋃4i=1 E i . It is not difficult to see that G is K2k+1 -free, and G → (4, 2k)v . Therefore, Fv (4, 2k; 2k + 1) ≤ 2Fv (3, k; k + 1) + 5k + 2 . It is not difficult to generalize Theorem 8.6.1 to more general cases.

8.6 Fv (3, k; k + 1) and other vertex Folkman numbers

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Based on Theorem 8.6.1 and the known upper bounds for Fv (3, k; k + 1), we can prove that Fv (4, 2k; 2k + 1) < 43 4 k + C1 holds for some constant C1 . For k that is large enough, this upper bound on Fv (4, 2k; 2k+1) is better than (23/2)k+C2 , which we can obtain based on Fv (4, 4; 5) ≤ 23. The upper bound on Fv (3, k; k + 1) was studied in [253]. We know that 2k + 4 ≤ Fv (3, k; k + 1) ≤ (23/8)k + C. We can see that the difference between the known upper and lower bounds for Fv (3, k; k + 1) is large. For the exact values, we know that Fv (3, 3; 4) = 14, Fv (3, 4; 5) = 13 and Fv (3, 5; 6) = 16. For the upper bounds, we know that Fv (3, 6; 7) ≤ 18, Fv (3, 7; 8) ≤ 22 and Fv (3, 8; 9) ≤ 23. Based on these results, Fv (3, 9; 10) ≤ 32, Fv (3, 10; 11) ≤ 31, Fv (3, 11; 12) ≤ 37, Fv (3, 12; 13) ≤ 36, Fv (3, 13; 14) ≤ 39, Fv (3, 14; 15) ≤ 41, and Fv (3, 15; 16) ≤ 45 were obtained in [253]. Based on these results, a theorem on the upper bound for Fv (3, k; k + 1) was proved in [253]. We can improve Fv (3, 9; 10) ≤ 32 and Fv (3, 11; 12) ≤ 37 in [253] easily as follows. Fv (3, 9; 10) ≤ Fv (3, 4; 5) + Fv (3, 5; 6) ≤ 13 + 16 = 29 , Fv (3, 11; 12) ≤ Fv (3, 5; 6) + Fv (3, 6; 7) ≤ 16 + 18 = 34 . Therefore, we have Fv (3, 9; 10) ≤ 29 and Fv (3, 11; 12) ≤ 34. Based on these, we can improve the theorem on the upper bound for Fv (3, k; k + 1) in [253] in the cases k ≡ 1 (mod 8) and k ≡ 3 (mod 8). We write the improved theorem as follows. Theorem 8.6.2. If k ≥ 5, then 23 { 8 k, { { { 23k+25 { , { { 8 { { { 23k+18 { , { 8 { { { 23k+19 { { , Fv (3, k; k + 1) ≤ { 8 23k+12 { , { { 8 { { { 23k+13 { , { { { 8 { 23k+6 { { , { 8 { { 23k+15 { 8 ,

k ≡ 0 (mod 8) ; k ≡ 1 (mod 8) ; k ≡ 2 (mod 8) ; k ≡ 3 (mod 8) ; k ≡ 4 (mod 8) ; k ≡ 5 (mod 8) ; k ≡ 6 (mod 8) ; k ≡ 7 (mod 8) .

We have no idea how to prove the following conjecture, which is similar to Conjecture 8.5.1, which states that Fv (k, k; k + 1) ≤ Fv (k + 1, k + 1; k + 2). Conjecture 8.6.2. If k is an integer and k ≥ 4, then Fv (3, k; k + 1) ≤ Fv (3, k + 1; k + 2) .

92 | 8 Folkman numbers It may not be easy to improve the known upper bounds on small Fv (3, k; k + 1) and to determine the exact values may be much more difficult because computing lower bounds may be very difficult. More generally than Conjecture 8.6.2, we may ask if Fv (s, k; k + 1) ≤ Fv (s, k + 1; k + 2) holds if k ≥ s + 1. Note that Fv (3, 3; 4) = 14 > 13 = Fv (3, 4; 5). Maybe the inequality above does not hold in some cases, but it holds for any k large enough. To improve the best known upper bound for Fv (3, k; k + 1) in Theorem 8.6.2, it is important to improve the upper bound for Fv (3, 8; 9), if the known upper bound 23 is not the exact value of Fv (3, 8; 9). We may consider whether Fv (3, 16; 17) ≤ 45 holds. If it holds, then we can improve the known upper bound for Fv (3, k; k + 1) in Theorem 8.6.2. Problem 8.6.1. Study the exact value and bounds for Fv (3, 8; 9). For the lower bound, we know that Fv (3, 8; 9) ≥ 20, which can be obtained by Fv (3, k; k + 1) ≥ 2k + 4. If this lower bound is not the exact value, we may try to improve it by computing. Note that Fv (2, 8; 9) = 17. For any graph G ∈ F(3, 8; 9; 20), if α(G) ≥ 4, then any subgraph of G obtained by deleting an independent set of order 4 must arrow (2, 8; 9)v , but its order is 16 and smaller than Fv (2, 8; 9), which is a contradiction. Therefore, α(G) ≤ 3. It is interesting to know whether there is a (9, 3)-graph in F(3, 8; 9; 20). We may also try to improve the known upper bound for Fv (3, k; k + 1) by finding an upper bound for Fv (3, k 0 ; k 0 + 1) smaller than (23/8)k 0 for another k 0 . Let us propose the following problem on the difference between Fv (4, k; k + 1) and Fv (3, k; k + 1). Of course, we can propose more similar problems that are more difficult. Let us begin with this problem that may be easier. Problem 8.6.2. Is there a positive constant C such that Fv (4, k; k + 1) − Fv (3, k; k + 1) ≤ C for any positive integer k? It is difficult to know whether there is such a constant C. We may also consider whether there is a positive constant C(k 0 ) such that Fv (k 0 + 1, k; k + 1) − Fv (k 0 , k; k + 1) ≤ C(k 0 ) , for given k 0 . We do not know the answer even for k 0 = 2. We know that Fv (3, k; k +1) ≥ 2k + 4 and Fv (2, k; k + 1) = 2k + 1. However, the known upper bound for Fv (3, k; k + 1) is much larger.

8.7 Universal graphs |

93

8.7 Universal graphs Universal graphs have been studied by Fan Chung, Graham, and other mathematicians (see,for instance, [60]). We may use universal graphs in computing lower bounds for some small vertex or edge Folkman numbers. Take, for example, the lower bound for Fe (3, 3; 4). In particular, we should consider the lower bounds for those small edge Folkman numbers for which the known lower bounds are small. Although we may obtain new lower bounds for Folkman numbers in only a very few cases, the method discussed in this section may be interesting in another way. In fact, we may have to deal with many small graphs in computing the lower bound for a small vertex or edge Folkman number. Universal graphs constructed in a related computation may be not enough to make the computation feasible to obtain a new lower bound. However, in some cases, some universal graphs constructed may be useful in computing upper bounds. That is to say, during the computation of lower bounds for a small Folkman number, we can obtain many graphs that may show us interesting upper bounds. Let G1 ∈ Fv (3, 3; 4; 14). Suppose that there are x graphs in Fv (3, 3; 4; 15) that contain G1 as an induced subgraph. Using these x graphs in Fv (3, 3; 4; 15), we can construct a large K4 -free graph G2 of order 14 + x that contains these x graphs in Fv (3, 3; 4; 15) as induced subgraphs. It may be possible to construct such a graph that contains these x graphs in Fv (3, 3; 4; 15) as induced subgraphs, of which the order is smaller than 14 + x. Suppose that G is any K4 -free graph of order 20 that arrows (3, 3)e and that G2 constructed above has subgraph that is isomorphic to the subgraph of G obtained by deleting an independent set of order 5. Then we can construct the possible graphs that may be isomorphic to G based on G2 . This is similar to what was done in proving Fe (3, 3; 4) ≥ 19 in [233], and the main difference is that these graphs may be much larger. It is not difficult to see that each of those graphs of order 20 is a subgraph of some graph among these large graphs. This idea may make the computation more difficult, if we only want to improve the lower bound on Fe (3, 3; 4) to 21. However, as discussed above, it is interesting because we may obtain a new upper bound for Fe (3, 3; 4) this way. This idea can be used to search for G ∈ Fe (3, 3; 4; 20) such that α(G) ≥ 5. Those graphs in Fe (3, 3; 4; 20) with independence number 4 can be discussed similarly. If we study K4 -free graphs on 21 vertices when we are studying the lower bound for Fe (3, 3; 4), we may follow this idea; otherwise there are too many graphs to deal with, which makes it unfeasible for us even though the graphs are small. We can consider the lower bound for Fv (4, 4; 5) by similar methods. Note that the known lower bound for Fv (4, 4; 5) is 17, which may be weak.

94 | 8 Folkman numbers

8.8 On the connectivity of Folkman graphs It is not difficult to prove some results on the connectivity of minimum Folkman graphs. Theorem 8.8.1. If s ≤ k, G → (s, k; k + 1)v and |V(G)| = Fv (s, k; k + 1), then the connectivity of G is no smaller than 2. Proof. It is clear that G must be connected, because otherwise one of its connected branches arrows (s, k)v , which contradicts |V(G)| = Fv (s, k; k + 1). Suppose that v ∈ V(G), and G −{v} = V1 ∪ V2 , and there is no edge between V1 and V2 , where V1 , V2 are not empty, and V1 ∩ V2 = 0. Suppose that v1 ∈ V1 and v2 ∈ V2 , and both vv1 and vv2 are in E(G). We can find v1 and v2 because G is connected and v is a cut vertex. Now we construct a graph H of order Fv (s, k; k + 1) − 1. Let V(H) = {v3 } ∪ V(G) − {v1 , v2 }. That is to say, we add a new vertex v3 to V(G) and delete v1 and v2 . For any u ∈ V(G)−{v1 , v2 }, let E1 = {v3 u | u ∈ V1 −{v1 }, v1 u ∈ E(G)}∪{v3 u | u ∈ V2 −{v2 }, v2 u ∈ E(G)}. Let E(H) = E(G − {v1 , v2 }) ∪ E1 ∪ {vv3 }. Note that |V(H)| = Fv (s, k; k + 1) − 1. We can see that the subgraph of G induced by V i ∪ {v} is isomorphic to the subgraph of H induced by {v, v3 } ∪ V i − {v i } for any i ∈ {1, 2}. It is not difficult to see that H → (s, k)v , which is a contradiction. Therefore, G is 2-connected. For edge Folkman numbers, we have the following two theorems. We omit the proofs here. Theorem 8.8.2. If G → (k, k; k + 1)e , and |V(G)| = Fe (k, k; k + 1), then the connectivity of G is not smaller than 3. Theorem 8.8.3. If s < k, G → (s, k; k + 1)e , and |V(G)| = Fe (s, k; k + 1), then the connectivity of G is not smaller than 2.

9 The Erdős–Hajnal conjecture In this chapter, we will discuss the Erdős–Hajnal conjecture.

9.1 The Erdős–Hajnal conjecture and its multicolor generalization The following Erdős–Hajnal conjecture was proposed in [104]. Conjecture 9.1.1. For every fixed graph H, there exists a constant f(H), so that every graph G without an induced subgraph isomorphic to H contains either a clique or an independent set of size |V(G)|f(H) . There are many references on the Erdős–Hajnal conjecture. For instance, in [13], Noga Alon, János Pach, and József Solymosi proved the Erdős–Hajnal conjecture for a special class of graphs and gave an equivalent reformulation for tournaments. There have been some studies on large cliques or independent sets in graphs without small path and antipath, for instance, [57], the results in which were improved later. Erdős and Hajnal also proposed studying a multicolor generalization of their conjecture. This states that for every fixed k-coloring of the edges of χ of a complete graph, there is an ϵ = ϵ(χ) > 0 such that every k-coloring of the edges of the complete graph on n vertices without a copy of χ contains a complete subgraph of order n ϵ , which only uses k − 1 colors (see [123]). In [123], Fox, Grinshpun, and Pach proved a weaker 1/2 estimate, replacing n ϵ by e ϵ(log n) . Note that the case of two colors is Conjecture 9.1.1.

9.2 The Erdős–Hajnal number The Erdős–Hajnal conjecture has been proved for some graphs. Our knowledge here is still quite limited. In particular, Lovász suggested the following very special case, which remains open. Problem 9.2.1. Is the Erdős–Hajnal conjecture true when H is isomorphic to C5 ? Zhao et al. [317] defined the Lovász–Ramsey number R C5 (k, l) as the smallest integer n such that every graph G of order n must contain a k clique, an l independent set, or an induced subgraph 5-cycle. Similarly, we define the Erdős–Hajnal number R H (k, l) as the smallest integer n: for every graph G of order n, G must contain a k clique, or an l independent set, or an induced subgraph H. In other words, the Erdős–Hajnal number R H (k, l) is the smallest integer n such that every (k, l)-graph of order n must contain an induced subgraph that

https://doi.org/10.1515/9783110576702-009

96 | 9 The Erdős–Hajnal conjecture is isomorphic to H. Therefore, the Lovász–Ramsey number R C5 (k, l) is a special case of the Erdős–Hajnal number R H (k, l), where H = C5 . In [317], some methods of studying classical Ramsey numbers were used to obtain bounds for R C5 (k, l), and some inequalities on different Lovász–Ramsey numbers were proved by constructive methods. In [317], it was proved that R C5 (k, l) ≤ R C5 (k − 1, l) + R C5 (k, l − 1) . For R H (k, l), we similarly have the following inequality: R H (k, l) ≤ R H (k − 1, l) + R H (k, l − 1) . In [317], it was proved that R C5 (3, s + t − 1) ≥ R C5 (3, s) + R C5 (3, t) − 1 . For R H (k, l), we can prove the following theorem without difficulty. Theorem 9.2.1. For any integers k ≥ 2, s ≥ 2 and t ≥ 2, if H is a connected graph of which the order is no smaller than 4, and H is also connected, then R H (k, s + t − 1) ≥ R H (k, s) + R H (k, t) − 1 . In [317], it was proved that R C5 (k, l) ≥ (k − 1)(l − 1) + 1 . For R H (k, l), we can prove the following theorem without difficulty. Theorem 9.2.2. Suppose that graph G is not a complete r-partite graph for any positive integer r. For any integers k ≥ 3 and l ≥ 3, we have R H (k, l) ≥ (k − 1)(l − 1) + 1 . It is interesting to determine the order of R C5 (3, l), which may be much smaller than R(3, l) and also smaller than R(C5 , K l ). Similar to the results on R(3, s) in [318], we have R C5 (3, s + t) − 1 ≥ R C5 (3, s + 1) − 1 + R C5 (3, t + 1) − 1 − 2 , and therefore, R C5 (3, s + t) ≥ R C5 (3, s + 1) + R C5 (3, t + 1) − 3 . We omit the details here. In [317], it was proved that R C5 (3, 4) = 8, R C5 (3, 5) = 10, R C5 (3, 6) = 13 and R C5 (4, 4) = 12. We list some lower bounds obtained in the following theorem [317].

9.2 The Erdős–Hajnal number

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Theorem 9.2.3. R C5 (4, 5) ≥ 15, R C5 (4, 6) ≥ 19, R C5 (5, 5) ≥ 19, R C5 (5, 6) ≥ 25. Problem 9.2.2. Improve the lower bounds or determine the exact values of R C5 (4, 5), R C5 (4, 6), R C5 (5, 5) and R C5 (5, 6). It is interesting to know whether R C5 (3, s + 1) < Fv (3, s; s + 1). If we can prove that R C5 (3, i + 1) < Fv (3, i; s + 1) holds for more i < s, then we may study the lower bound for Fv (3, s; s + 1) by this result. An algorithm for computing the lower bound on small R H (k, l) is to search for large subgraphs of a (k, l)-graph that contains no induced H. In particular, we may do some computing for the case H = C5 . If we compute the lower bound for small R H (k, k), we may let the (k, k)-graph considered be some Paley graph. It is not difficult to obtain some small results on R C5 (k, l). Note that R C5 (3, k) ≤ R(C5 , K k ). In [278] and [182], it was proved independently that R(C2k+1 , K n ) ≤ c(k)

n1+1/k , (ln n)1/k

for every fixed positive integer k, and n trends to infinity. We know that R(C5 , K8 ) is between 29 and 33, and R(C2a+1 , K k ) = 2a(k − 1) + 1 holds for any other case in which a ∈ {1, 2, 3, 4} and k ∈ {3, 4, 5, 6, 7}. One may conjecture that R(C2a+1 , K k ) = 2a(k − 1) + 1 for a ≥ 2 and k ≥ 2a; in particular, one may conjecture that R(C 5 , K k ) = 4k − 3 for k ≥ 8. However, this is not true. In fact, it was proved in [39] that for fixed l ≥ 4 and k → ∞, the cycle-complete Ramsey number satisfies R(C l , K k ) = Ω((k/(log k))(l−1)/(l−2) ). Therefore, we can see that 4 3 k R(C5 , K k ) = Ω (( ) ) . log k This is better than the upper bound for R(C5 , K k ) obtained in [278] and [182] cited above. A good lower bound on R C5 (3, s + 1) − R C5 (3, s) is interesting, yet may be not easy to prove. Define R(G1,1 , . . . , G1,j1 ; . . . ; G s,1 , . . . , G s,j s ) to be the minimum positive integer n such that if we color all the edges in K n with s colors, then there is a subgraph in color i that is isomorphic to G i,j for some i ∈ {1, . . . , s} and j ∈ {1, . . . , j i }. Therefore, R C5 (3, l) = R(C3 , C5 ; K l ). Now, let us propose a problem on the Erdős–Hajnal number R H (k, l). Problem 9.2.3. Study the bounds for the Erdős–Hajnal number R H (k, l), in particular, R C5 (k, l). It is not difficult to prove that R K k −e (k, l) = R(K k − e, K l ). Further, it is easy to prove that R H (k, k) = R H (k, k). When k < l, if H has fewer edges than H, then R H (k, l) < R H (k, l) may hold with a high probability. More computing and a detailed analysis will be interesting and may be not easy. We propose the following problem.

98 | 9 The Erdős–Hajnal conjecture Problem 9.2.4. Suppose that k and l are integers and k > l. Is there a connected graph H of order n, such that H is connected, |E(H)| < (1/2)C2n , and R H (k, l) < R H (k, l)? Let h(G) = hom(G) = max{cl(G), α(G)}. In [124] the following result was proved. Theorem 9.2.4. There are positive constants c3 and c4 such that for all n, k, every graph 1/2 on n vertices is k-universal or satisfies hom(G) ≥ c3 2c 4((log n)/k) log n. Now let us propose a Erdős–Hajnal type conjecture related to universal graphs. Conjecture 9.2.1. Given positive integer k and any ϵ > 0, there is an integer n such that if G is any (a, a)-graph of order n, and H is any graph of order k, then G contains an induced subgraph that is isomorphic to H, where a = ⌊n ϵ ⌋. That is, G is k-universal. For given n ≥ t ≥ 4, let f(H, n) = min{h(G) | |V(G)| = n, G is H-free} , and f t (n) = max{f(H, n) | |V(H)| = t} . Problem 9.2.5. Study which graph H of order t may achieve the maximum for f t (n). Maybe the size of such H is about t(t − 1)/4. So, considering the case in which the order of H is small may help for us to get a deeper understanding of this problem. Let us consider R H (k, l) in which |V(H)| is not small related to R(k, l). If H is a maximal (k, l)-graph, and |V(H)| < R(k, l) − 1, then R H (k, l) = R(k, l). For instance, K k−1 [E l−1 ] is a maximal (k, l)-graph. If |V(H)| = R(k, l) − 1 and there is a (k, l)-Ramsey graph that is not isomorphic to H, then R H (k, l) = R(k, l). Otherwise, if H is the unique (k, l)-Ramsey graph, then R H (k, l) = R(k, l) − 1. In general, R H (k, l) ≤ R(k, l); R H (k, l) < R(k, l) if and only if every (k, l)-Ramsey graph contains an induced graph that is isomorphic to H.

9.3 On the lower bound on the vertex Folkman number F v (k, k; k + 1) Let G be a (3, k)-graph. Therefore, G is C5 -free if and only if it is induced C5 -free. It is interesting to study how many vertices G can have. Now let us prove the following theorem. Theorem 9.3.1. Let G be a (3, k)-graph. If G is C5 -free, and v is any vertex in G, then {u | d(u, v) = 2} ∪ {v} is an independent set. Proof. If the degree of v in graph G is 1, then it is not difficult to see that the theorem holds. Now let us suppose that the degree of v in graph G is d(v) ≥ 2, and the set of its neighbors is {u 1 , . . . , u d(v)}. Note that this is an independent set because G is K3 -free.

9.4 More generalizations

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Any vertices u and u 󸀠 such that d(u, v) = 2 and d(u 󸀠 , v) = 2 must be a neighbor of at least one vertex in {u 1 , . . . , u d(v) }. If u and u 󸀠 have common neighbors in graph G, then there is no edge between them. Now suppose that they have no common neighbors, and uu i and u 󸀠 u j are edges in G, where i ≠ j. Therefore, there is no edge between u and u 󸀠 because G is C5 -free. So, {u | d(u, v) = 2} is an independent set. Since in this independent set there is no neighbor of v, the theorem holds. Based on this theorem we can obtain the following result, which may be used in studying the lower bound for Fv (k, k; k + 1). Theorem 9.3.2. Let G be a (3, k)-graph. If υ(G) > 2(k − 1) and G is C 5 -free, then there are two vertices u and v in G, such that u is not a neighbor of v, and they have no common neighbors. It is not difficult to see that for such a graph G in Theorem 9.3.2, if we add an edge between vertices u and v, we obtain a new (3, k)-graph. Hence, δ(G) < k − 1. So, the complement of graph G does not arrow (k − 1, k − 1)v , and based on all this we can obtain Fv (k, k; k + 1) ≥ 5 + Fv (k − 2, k; k + 1). For k ≥ 4, it will be interesting to see if we can prove that if integer k ≥ 4, then Fv (i + 2, k; k + 1) ≥ Fv (i, k; k + 1) + 5, where 2 ≤ i ≤ k − 2.

9.4 More generalizations Similarly to the Erdős–Hajnal number R H (k, l), we may define R A (k, l) for a forbidden graph set A = {G i | 1 ≤ i ≤ m}. For a graph set A = {G i | 1 ≤ i ≤ m}, define R A (k, l) as the smallest integer n: for every (k, l)-graph G of order n, G must contain an induced subgraph H that is isomorphic to some graph in A. Recall that a graph is k-universal if it contains all graphs on k vertices as induced subgraphs. We may also define the following R󸀠A (k, l). For a graph set A = {G i | 1 ≤ i ≤ m}, define R󸀠A (k, l) as the smallest integer n: for every (k, l)-graph G of order n, and any graph H1 in A, there is an induced subgraph H in G isomorphic to H1 . It is easy to see that R A (k, l) ≤ R󸀠A (k, l) .

10 Other Ramsey-type problems in graph theory There are numerous variations of Ramsey numbers or related topics. We will deal with some of them in this chapter. Other ones like zero-sum Ramsey numbers, irredundant Ramsey numbers, chromatic Ramsey numbers, or avoiding sets of graphs in some colors, will not be considered. Note that these problems may relate to some topics discussed in this book and the interested reader can consult related references without difficulty.

10.1 Ramsey-type numbers based on directed graphs We found and read the problems in the first subsection below long ago. We found those problems in the second subsection much later. We believe that these topics are natural and should be studied more in the future.

10.1.1 Directed Ramsey numbers for acyclic subtournaments Let R(k) denote the smallest positive integer n such that every tournament with n players (i.e., the complete directed graph on n nodes) contains an acyclic k-player subtournament. Re-expressed in voting-theory language: R(k) is the smallest number n of candidates in a ranked-ballot election, such that it is guaranteed that at least one subset containing at least k of those candidates can be unambiguously ranked in relation to one another. Let TT k be the transitive tournament of order k. In most cases, it is difficult to find the maximum transitive subtournaments. There are some references on computing the maximum transitive subtournaments; see, for instance, [168]. Transitive tournaments play a role in Ramsey theory analogous to that of cliques in undirected graphs. Similarly to the classical Ramsey number R(k, k), we know that every tournament on n vertices contains a transitive subtournament on 1 + ⌊log2 n⌋ vertices [276]. The proof is simple: choose any vertex v to be part of this subtournament and form the rest of the subtournament recursively on either the set of incoming neighbors of v or the set of outgoing neighbors of v, whichever is larger. Therefore, R(k) ≤ 2R(k − 1). For instance, every tournament on seven vertices contains a three-vertex transitive subtournament; G7 , the Paley tournament on seven vertices shows that this is the most that can be guaranteed ([107]), because G7 does not contain TT4 . In [107] it was conjectured that for each positive integer k there exists a TT k -free tournament of order 2k−1 − 1. However, Reid and Parker disproved this conjecture in [238]. In fact, the conjecture does not hold even for k = 5, because R(5) = 14. https://doi.org/10.1515/9783110576702-010

102 | 10 Other Ramsey-type problems in graph theory

In [107] it was proved that there are tournaments on n vertices without a transitive subtournament of size 2 + ⌊2 log2 n⌋. The proof uses a counting argument. Therefore, if n = R(k), we have 1 + ⌊log2 n⌋ ≤ k ≤ 1 + 2⌊log2 n⌋ . How to improve the bounds for R(k) is the main problem in this section. Problem 10.1.1. Improve the known lower and upper bounds for R(k). It is not difficult to prove that R(k) ≤ R(k, k). For any tournament T = (V, A) with the n-element vertex set V and n = R(k, k), let us color the edges in K n as follows and let the edge-colored K n be G, where V(G) = {u 1 , . . . , u n }. If i < j and v j v i ∈ A, then color u i u j with red; if i < j and v i v j ∈ A, then color u i u j with blue. Therefore, there is a monochromatic K k in G. Hence, we can find a TT k in T. Therefore, R(k) ≤ R(k, k). Problem 10.1.2. Prove or disprove R(k) < R(k, k). It is not difficult to know that R(k) < R(k, k) holds for any k ∈ {3, . . . , 11}. We can see that R(12) ≤ 1728 and we know that R(12, 12) ≥ 1639, and we do not know if R(12) < R(12, 12) holds. We know the values of R(k) for every positive integer k ≤ 6. These are R(1) = 1, R(2) = 2, R(3) = 4, R(4) = 8, R(5) = 14 and R(6) = 28. The simplest open case is R(7), and 32 ≤ R(7) ≤ 54. As we can see, the gap between the upper and the lower bounds is not small. Maybe the lower bound is weak, but there seems no reference on it. In general, R(k) ≤ 54 ⋅ 2k−7 = 27 ⋅ 2k−6 for k ≥ 8. There are some known results on the lower bound for more small R(k). Most of these lower bounds were obtained by directed Paley graphs, of which the order p is a prime and p ≡ 3 (mod 4). Note that for odd prime p, −1 is not a quadratic residue of p when p ≡ 3 (mod 4). More general than Paley tournaments, we may consider the cyclic tournaments to study R(k) for small k. For two tournaments D1 and D2 , we can define D1 [D2 ] similarly to G[H], where G and H are two graphs. Suppose that D1 is TT a -free and D2 is TT b -free, and |D1 | = R(a) − 1, |D2 | = R(b) − 1. Therefore, by D1 [D2 ] it is not difficult to prove that R((a − 1)(b − 1) + 1) ≥ (R(a) − 1)(R(b) − 1) + 1 , in particular, if b = 3, then we have that R(2a − 1) ≥ 3R(a) − 2 by R(3) = 4. This is similar to R(a1 b 1 + 1, a2 b 2 + 1) − 1 ≥ (R(a1 + 1, a2 + 1) − 1)(R(b 1 + 1, b 2 + 1) − 1) . Both seem weak. It is easy to see that R(k+1)−R(k) ≥ 2. We may consider the following problem, which may be very difficult. Problem 10.1.3. Does R(k + 1) − R(k) tend to infinity when k tends to infinity?

10.1 Ramsey-type numbers based on directed graphs |

103

10.1.2 Directed Ramsey numbers The directed Ramsey number r(D1 , . . . , D k ) was defined in [29] in 1974. There are some papers on r(D1 , . . . , D k ), for instance, [193]. The following two paragraphs are on the basic definition and basic facts on the directed Ramsey number r(D1 , . . . , D k ), and we follow [193] here. Let D1 , . . . , D k be acyclic directed graphs (some or all may be identical). We define r(D1 , . . . , D k ) as the largest integer n for which there exists a tournament T = (V, A) with an n-element vertex set V and with a k-coloring φ : A → {1, . . . , k} on its arc set A such that no D i is a subdigraph of T in color i for any i ∈ [k]. In order to ensure n < ∞ in the definition of the directed Ramsey number r(D1 , . . . , D k ), it is necessary to assume that each D i is acyclic, for otherwise the transitive tournaments colored completely with color i provide arbitrarily large admissible constructions. On the other hand, if each D i is acyclic, the basic theorem of Ramsey theory together with the simple fact r(TT s ) ≤ 2s−1 (for all s > 0) yields n < ∞, where s = max{|V(D i )| | 1 ≤ i ≤ k}. Note that if we consider the case k = 1, then r(TT s ) is R(s) − 1 in last subsection. So, r(D1 , . . . , D k ) + 1 is a generalization of R(s). Maybe there will be more work done on directed Ramsey numbers in the future. Problem 10.1.4. Study the lower and upper bounds for r(TT s , TT k ), where s ≥ k ≥ 3.

10.1.3 Directed Folkman numbers Suppose that s and k are positive integers, and s > k ≥ 3. Let the directed Folkman number Fd (k; s) be the smallest positive integer n for which there is a K s -free graph G such that any directed graph based on G contains TT k . It is not difficult to prove the existence of Fd (k; s) based on the existence of Fe (k, k; s). Theorem 10.1.1. If s and k are positive integers and s > k ≥ 3, then Fd (k; s) exists and Fd (k; s) ≤ Fe (k, k; s). Proof. Suppose that G is a K s -free graph such that G → (k, k)e , and V(G) = {v1 , . . . , v n }, where n = Fe (k, k; s). For any directed graph D based on G, if i < j, we color edge v i v j ∈ E(G) with red if there is arc v i v j in D, and color edge v i v j ∈ E(G) with blue if there is arc v j v i in D. Because G → (k, k)e , we can see that there must be a monochromatic K k in such a red-blue coloring of E(G), and therefore there must be a transitive subtournament TT k in D. Thus, Fd (k; s) exists and Fd (k; s) ≤ Fe (k, k; s). Let us consider the exact value of Fd (3; 4). We have Fd (3; 4) > 5 because the chromatic number of every K4 -free graph of order 5 is no larger than 3. Note that if Fd (k; s) = n, then there must be a K s -free graph G such that χ(G) ≥ R(k). On the other hand, Fd (3; 4) ≤ 6 can be proved by K1 + C5 . In fact, in any directed graph based on K1 + C5 , the unique vertex of degree 5 must be in at least one TT3 . Therefore, Fd (3; 4) = 6. We

104 | 10 Other Ramsey-type problems in graph theory can see that Fd (3; 4) is much smaller than 20, the known lower bound for Fe (3, 3; 4). This fact leads us to propose the following problem. Problem 10.1.5. Prove or disprove Fd (k; s) < Fe (k, k; s), where s > k ≥ 3. The main problem in this section is the following problem. Problem 10.1.6. Study the lower and upper bounds for Fd (k; k + 1), including Fd (4; 5) as the smallest unknown case. Let us discuss Fd (4; 5) a little more. Some related ideas may be used in dealing with Fd (k; k + 1) for k > 4 similarly. If H → (K1 + C5 , K1 + C5 ; K4 )v , then we can obtain an upper bound on Fd (4; 5) based on K1 + H. Note that any directed graph based on K1 + C5 contains TT3 . That is, Fd (4; 5) ≤ 1+ Fv (K1 + C5 , K1 + C5 ; K4 ). So it is interesting to obtain an interesting upper bound for Fv (K1 + C5 , K1 + C5 ; K4 ), which is not easy without doing some computing. On the other hand, Fd (4; 5) may be much smaller than Fv (K1 + C5 , K1 + C5 ; K4 ), and studying the upper bound for Fd (4; 5) based on some small K5 -free graph may be a better idea.

10.2 Hypergraph Ramsey numbers As we know, a hypergraph is a generalization of graph. If all edges have the same cardinality k, the hypergraph is said to be uniform or k-uniform, or is called a k-hypergraph. Most works on hypergraph Ramsey numbers studied uniform hypergraphs. For instance, in [173], hypergraph Ramsey numbers for uniform hypergraphs were studied. Suppose that k, s, n are positive integers and k ≤ s, k ≤ n. The hypergraph Ramsey number r k (s, n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all k-tuples from this set are red (blue). By Ramsey’s theorem we know that r k (s, n) exists. We can find a survey and some new results on hypergraph Ramsey numbers in [71]. A more detailed survey can be found in [229]. Some known results on small hypergraph Ramsey numbers can be found in [229]. For instance, in [195], r3 (4, 4) was determined to be 13, and Exoo [117] obtained a lower bound 82 for r3 (5, 5). The construction that shows r3 (4, 5) ≥ 33, obtained by Exoo, is available at http://ginger.indstate.edu/ge/ RAMSEY. Song Enmin proved some constructive results on hypergraph Ramsey numbers in [272]; the following results were proved. r4 (p, q) ≥ 2r4 (p − 1, q) − 1 for p, q > 4 , and r4 (p, q) ≥ (p − 1)r4 (p − 1, q) − p + 2 for p ≥ 5, q ≥ 7 .

10.2 Hypergraph Ramsey numbers

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Erdős, Hajnal, and Rado [105] showed that there are positive constants c and c󸀠 such 2 c󸀠 n that 2cn < r3 (n, n) < 22 . They proposed the following conjecture, and Erdős offered a 500 dollars reward for a proof. Conjecture 10.2.1. Hypergraph Ramsey number r3 (n, n) ≥ 22 c > 0.

cn

for some constant

That is, the authors of [105] believed that the known upper bound for r3 (n, n) is much better than the known lower bound. We believe that it will be better if we can determine the constant rather than only proving its existence. Of course, both of these are far from being reached. A triple system or 3-graph H with vertex set V(H) is a collection of 3-element subsets of V(H). Write K 3n for the complete 3-graph with vertex set of size n. Given 3-graphs F, G, the Ramsey number r(F, G) = r3 (F, G) is the minimum n such that every red/blue coloring of K 3n results in a monochromatic red copy of F or a monochromatic blue copy of G. Similarly, we may also consider multicolor generalized hypergraph Ramsey numbers. For instance, we know that 13 ≤ r3 (K4 − e, K4 − e, K4 − e) ≤ 16, where the lower bound was obtained in [114], and the upper bound is easy to obtain. We may write r3 (K4 − e, K4 − e, K4 − e) as R(K4 − e, K4 − e, K4 − e; 3) or R3 (K4 − e; 3) as done by some authors. In [19] it was proved that R k (3) ≤ R4k (K4 − e; 3) ≤ R4k (3) + 1 . There are several natural ways to define a cycle in hypergraphs. For s > 3, the tight cycle C3s is the 3-graph with vertex set ℤ s (integers modulo s) and edge set {{i, i + 1, i + 2} : i ∈ ℤ s }. Mubayi and Rödl proved the following theorem on r(C3s , K 3n ) [204]. Theorem 10.2.1. Fix s ≥ 5 and s ≡ 0 (mod 3). There are positive constants c1 and c2 such that 2 2c 1 n < r (C3s , K 3n ) < 2c 2 n log n . Mubayi improved the upper bound in the following theorem [203]. Theorem 10.2.2. Fix a positive integer s ≡ 0 (mod 3) such that s ≥ 16 or s ∈ {8, 11, 14}. There is a positive constant c s such that r (C3s , K 3n ) < 2c s n log n . It would be interesting to know whether the inequality in Theorem 10.2.2 holds in other small cases. Problem 10.2.1. For any s ∈ {4, 5, 6, 7, 9, 10, 12, 13, 15}, does r (C3s , K 3n ) < 2c s n log n ?

106 | 10 Other Ramsey-type problems in graph theory

If we can prove the inequality in Problem 10.2.1, the gap between the known lower and upper bounds, 2c 1 n and 2c s n log n , respectively, is still large. It is interesting to study the bounds for r(C3s , K 3n ) more. In [153], the generalization of Ramsey theorem for hypergraphs for local colorings was proved. We may consider the problem on hypergraphs similarly to Problem 15.1.1 on the local colorings of graphs. The question is, does the local hypergraph Ramsey number considered always equals the related hypergraph Ramsey number? Problem 10.2.2. Does the local hypergraph Ramsey number always equals the related hypergraph Ramsey number?

10.3 Size Ramsey numbers Denote by re (H, q) the size Ramsey number of H with respect to coloring with q colors. That is, re (H, q) = min{|E(G)| | G → (H)q }. The study of re (K n , q) is essentially equivalent to the study of the original Ramsey number. Namely, it can be verified that if re (K n , q) = m, then a complete graph with exactly m edges has the desired property. This result is attributed to Chvátal in [101]. So, we need only to consider the following general problem for size Ramsey numbers. Problem 10.3.1. Study the bounds and exact value of re (H, q), where H is a connected non-complete graph. ̂ H), which Some references can be found on the two-color size Ramsey number R(F, e ̂ H) = min{|E(G)| | G → (F, H) }. We can see that need not be diagonal. Here, R(F, ̂ R(H, H) = re (H, 2). We mention only the following problem here. ̂ n , P n ). ̂ n , P n ) and R(C Problem 10.3.2. Study the bounds on R(P There are some references on this problem, and now we know that 5 2n

̂ n , P n ) ≤ 74n , − O(1) ≤ R(P

for sufficiently large n [83]. In [82], it was proved that R(C n , P n ) ≤ 2257n , for even and sufficiently large n. Instead of working directly in the uniform probability space of random regular graphs, the pairing model (also known as the configuration model) of random regular graphs was used in [82].

10.5 Induced Ramsey numbers |

107

10.4 Ramsey multiplicities The Ramsey multiplicity M(G, n) of a graph G is the minimum number of monochromatic copies of G over all 2-coloring of the edges of K n (see [122]). An exact formula for M(K3 , n) was found by Goodman in 1959 in [139]. Let M(G) be M(G, R(G, G)). The study of Ramsey multiplicity M(G) was initiated in 1974 by Harary and Prins in [159]. We can find a survey in [55], in which many related results before 1980 were surveyed. It was proved that M(K4 ) = 9 in [221]. It is not difficult to define the multicolor Ramsey multiplicity M r (G) similarly to Ramsey multiplicities. Problem 10.4.1. The multicolor Ramsey multiplicity M r (G) of a graph G is defined as the smallest number of monochromatic copies of G in any r-coloring of edges of K R r (G) . Study the bounds for M r (G). Since we know the exact values of very few multicolor classical Ramsey numbers, we only can obtain a few interesting results for Problem 10.4.1 now. On the other hand, we may do some computation, which may help us to understand the problem better. For instance, we may try the following simple case first. Problem 10.4.2. Compute M3 (K3 ).

10.5 Induced Ramsey numbers The induced Ramsey number r∗ (G, H) is the smallest positive integer n such that there exists a graph F of order n such that any 2-edge-coloring (red and blue) of F yields an induced copy of G in red or an induced copy of H in blue. If H is isomorphic to G, then we denote r∗ (G, H) by r∗ (G). The existence of r∗ (G, H) was verified by Rödl in his doctoral thesis [242], and also verified independently by other mathematicians. Some general upper bounds on r∗ (G, H) for various graphs G and H can be found in [170]. It is not difficult to propose many conjectures on induced Ramsey numbers. The following conjecture on the induced Ramsey number was proposed by Erdős in [97]. Conjecture 10.5.1. The induced Ramsey number r∗ (G) < c n , where |V(G)| = n and c is some absolute constant. On the other hand, we may consider the induced Ramsey numbers with bounded clique numbers, either edge or vertex type, which may be regarded as a Folkman type problem. Note that Rödl and some other mathematicians did some interesting work on these topics.

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10.6 The bipartite Ramsey number Define the bi-color bipartite Ramsey number b(s, t) to be the smallest n such that in any red-blue coloring of the edges in K n,n , there is a monochromatic K s,s in red or a monochromatic K t,t in blue. The multicolor generalization of b(s, t), b(s1 , . . . , s m ), can be defined similarly. It is not difficult to find some references on the multicolor bipartite Ramsey number b(s1 , . . . , s m ) with s1 = ⋅ ⋅ ⋅ = s m = 2, b m (2), where all the bipartite graphs are isomorphic to C4 . For instance, we know that b(2, 2) = 5, b 3 (2) = 11, b 4 (2) = 19 and 26 ≤ b 5 (2) ≤ 28. It was conjectured [91] that b 5 (2) = 28. In [69], it was proved that b(k, k) ≤ (1 + o(1))2k+1 log k , where the log is taken to the base 2. On the other hand, the best known lower bound is k k+1 (1 + o(1)) 2 2 , e the proof for which is, like Spencer’s lower bound for R(k, k) in [273], based on the Local lemma. It is obvious that the gap between the known upper and lower bounds for b(k, k) is large. So, it is natural to propose the following problem. Problem 10.6.1. Study the upper and lower bounds for b(k, k). It is known that b(2, 2) = 5, b(2, 3) = 9 and b(3, 3) ≥ 15. For off-diagonal cases, even the following case is not solved. Problem 10.6.2. Compute the value of b(2, 2, 3). It is not difficult to propose some problems on bipartite Ramsey numbers similar to those for classical Ramsey numbers. We propose only the following one here. Problem 10.6.3. Study the value and bounds for b(k + 1, k + 1) − b(k, k). Suppose that b ≥ s. The Zarankiewicz number z(b; s), i.e., the bipartite Turán number, is the maximum size of a subgraph of K b,b , which does not contain K s,s as a subgraph. The Zarankiewicz number has been an important tool in studying bipartite Ramsey numbers. General Zarankiewicz numbers z(m, n; s, t) can be defined similarly to z(b; s). General Zarankiewicz numbers z(m, n; s, t) and related extremal graphs have been studied by numerous authors. It is known that 17 ≤ b(2, 2, 3) ≤ 18, and the bounds were found by Collins, Riasanovsky, Wallace, and Radziszowski based on their approach and knowledge on the Zarankiewicz number [68]. It is known that z(17; 2) = 74 and z(17; 3) ≤ 141, so we can see that 74 + 74 + 141 = 289, which is just barely too large. There is also some work on br(K s1 ,t 1 , K s2 ,t 2 ) that is defined to be the smallest n such that in any red-blue of the edges of K n,n , there is a monochromatic K s1 ,t 1 in red

10.6 The bipartite Ramsey number | 109

or a monochromatic K s2 ,t 2 in blue. The interested reader can find related references without difficulty. It is not difficult to generalize it to the multicolor case. Suppose that s1 ≤ t1 and s2 ≤ t2 . There may be integers a and b such that in any bi-coloring of the edges of K a,b , there is a monochromatic K s i ,t i for some i ∈ {1, 2}, where a + b < 2br(K s1 ,t 1 , K s2 ,t 2 ). We will not discuss this problem in more detail.

11 On van der Waerden numbers and Szemerédi’s theorem Additive combinatorics is a newish and active branch of mathematics that grew out of combinatorial number theory, with input from many other areas such as harmonic analysis, ergodic theory, analytic number theory, group theory, and extremal combinatorics (see [141]). There are many important results and open problems of Ramsey type in additive number theory. In this chapter, we will discuss some problems on sets without long arithmetic progressions among them. For an integer k ≥ 2, an arithmetic progression of length k is a sequence of the form a, a + d, . . . , a + (k − 1)d, where a, d are integers and d > 0. Throughout this chapter, “arithmetic progression of length k” is abbreviated by k-AP. A sequence (or set) of positive integers is k-AP-free if it contains no k-AP. Let r k (n) denote the size of the largest k-AP-free subset of [n]. Let k be a positive integer and 0 < δ < 1. Szemerédi’s theorem [285] asserts that there exists the smallest positive integer N(k, δ) such that every subset of [N] of size at least δN contains a k-AP, provided N ≥ N(k, δ). That is, Szemerédi’s theorem asserts that r k (n) = o(n) for each k, a conjecture proposed in [112] by Erdős and Turán in 1936. Furstenberg gave an ergodic theory proof of Szemerédi’s theorem in [128]. Gowers gave a new proof of Szemerédi’s theorem in [140], in which the best known general upper bound for N(k, δ) was obtained for δ ∈ (0, 1/2]. Let k 1 , . . . , k m and k be integers larger than 1. The van der Waerden number W(k 1 , . . . , k m ) is the smallest positive integer n such that every m-coloring of [n] contains a monochromatic k i -AP in color i, for some i ∈ [m]. When k 1 = ⋅ ⋅ ⋅ = k m = k, we write W(k 1 , . . . , k m ) as W m (k). In [289] van der Waerden proved that W m (k) exists for all integers m and k larger than 1. We can see that Szemerédi’s theorem implies the van der Waerden theorem, and W m (k) ≤ N(k, 1/m). There are polynomial generalizations of the van der Waerden theorem. We will not discuss them. We know that Ramsey proved his famous theorem during his study on a logic problem. Let us discuss the early history of the van der Waerden theorem now. We can find a well-written article including a survey of this history in “Complete disorder is impossible: The mathematical work of Walter Deuber” written by Prömel. We will cite related history from this article. Schur proved his famous theorem on sum-free sets in [249] in 1916. He also worked on the distribution of quadratic residues and non-residues modulo p for odd prime p, an old problem in number theory. He conjectured that for every positive integer k and every sufficiently large prime p there exist k consecutive integers, which are quadratic residues, and there exist k consecutive integers, which are quadratic nonhttps://doi.org/10.1515/9783110576702-011

112 | 11 On van der Waerden numbers and Szemerédi’s theorem

residues (modulo p). These problems are natural and interesting. Instead of considering them directly, Schur first tried to show that for every k there exists an n such that, for every coloring of [n] with two colors, one of the two color classes contains a k-AP. Schur did not succeed in any of these questions, and both remained open until van der Waerden and other mathematicians in Göttingen heard about the problem, and van der Waerden proved his famous theorem (developed through discussions with Artin and Schreier). Although N(k, δ) is not among Ramsey-type problems, it is important in the study of van der Waerden numbers because the best-known upper bound for W(k, k) was obtained by Gowers based on a general upper bound for N(k, 1/2) in [140]. The difference between W(k, k) and N(k, 1/2) may be very large, but no one knows how to obtain an upper bound for W(k, k) that is smaller than N(k, 1/2). The following conjecture proposed by Erdős is one of the most famous conjectures on arithmetic progressions. It is still open, even for 3-AP. Conjecture 11.1 ([99]). Any set A of positive integers whose sum of reciprocals ∑n∈A 1/n is divergent must contain arbitrarily long AP. Note that if we can prove this conjecture, then we obtain a new proof of the Green– Tao theorem that states that the primes contain arbitrarily long AP. In the proof of the Green–Tao theorem, not only the density but also the structure of the prime set were used. Before we discuss more problems related to van der Waerden numbers and Szemerédi’s theorem, let us survey some known results on r k (n) first.

11.1 Known bounds for r k (n) In this section, we will survey some known general results on lower and upper bounds for r k (n), where k ≥ 3. The exact size of r k (n) is still unknown. As we can see below, the gap between the known lower and upper bounds for r k (n) is very large. In general, for k ≥ 5, the best known upper bound is r k (n) ≤

n −2k+9

,

(log2 log2 n)2

due to Gowers ([140]), and for k = 4 the best known upper bound is r4 (n) ≤ C

n e c√log log n

for some C > 0 by Green and Tao [150]. For the upper bound on r3 (n), we know that log n) n r3 (n) ≤ C (loglog n 4

11.2 The van der Waerden number W(k, k)

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for some absolute constant C > 0, due to Bloom ([35]). It was proved by O’Bryant [214] that for every ϵ > 0, if n is sufficiently large, then 6 ⋅ 23/4 √5 1 − ϵ) exp (−√8 log n + log log n) , 3/2 4 eπ 1 t log log n) , r k (n) ≥ nC k exp (−t2(t−1)/2√log n + 2t

r3 (n) ≥ n (

where C k > 0 is an unspecified constant, log = log2 , exp(x) = 2x , and t = ⌈log k⌉. These are currently the best lower bounds for all k ≥ 3. In [214], it was also pointed out that r3 (n) ≥ n2−√8 log n . For k ≥ 1 + 2t−1 , there exists a positive constant c such that for all n ≥ 1 r k (n) ≥ cn

2t √ log n

. t (t−1)/2√log n 2t2 These lower bounds are currently the best constructive ones known.

11.2 The van der Waerden number W(k, k) In this section, let us consider some problems on van der Waerden numbers. The following problem was proposed by R. Graham, who offered 1000 USD for its proof. 2

Conjecture 11.2.1. W(k, k) ≤ 2k for any integer k > 2. The best known upper bound for W(k, k) is based on the best known upper bound for N(k, 1/2) in [140]. It is 22

W(k, k) ≤ 22

2k+9

.

We know that W(3, 3) = 9, W(4, 4) = 35, W(5, 5) = 178, and W(6, 6) = 1132. For W(7, 7), the best known lower bound is 3704, and no upper bound better than that implied by the general upper bound by Gowers is known. Problem 11.2.1. Give the exact value or interesting bounds for W(7, 7). The following problem is one of the most important problems on van der Waerden numbers. Problem 11.2.2. W(k, k) < N(k, 1/2) for any integer k ≥ 3? In small cases, we know that W(k, k) < N(k, 1/2) holds for any k ∈ {3, 4, 5} by known data. It is not difficult to prove the following simple theorem. Theorem 11.2.1. For m ≥ 2, if m | W m (k), then W m (k) < N(k, 1/m). We know that W(6, 6) = 1132, therefore, there exists 6-AP-free subset of order 566 in [1132], and N(6, 1/2) > 1132. Hence, W(k, k) < N(k, 1/2) holds when k = 6 too.

114 | 11 On van der Waerden numbers and Szemerédi’s theorem By computing, it is not difficult to know that r7 (2401 × 6) ≥ 1296 × 6. We can see that 1296/2401 > 1/2, hence, N(7, 1/2) > 1296×6×2 = 15,552. This lower bound for N(7, 1/2) is much larger than 3704, the best known lower bound for W(7, 7). Note that we know that W(8, 8) > 11,495. Although we cannot prove that N(7, 1/2) > W(7, 7), it seems that it holds with a high probability. Let us discuss a conjecture of Dan S. Hendrick and Mridul Mehta on W(k, k). In a preprint written in 1996 titled “van der Waerden numbers”, Hendrick and Mehta proposed some conjectures on van der Waerden numbers. Based on these conjectures, they predicted that W(k, k) ∼ M k (k − 1) and M k+1 ∼ kM k . Based on M5 = 44 they conjectured that W(6, 6) is about 1100 when the known lower bound for W(6, 6) is 696. Note that we know W(6, 6) = 1132 now [198]. By what they conjectured and some simple computation, we can see that what they conjectured is W(k, k) ∼ (11/6)(k − 1)(k − 1)! for k ≥ 6. We list this as a conjecture. Conjecture 11.2.2. W(k, k) ∼ (11/6)(k − 1)(k − 1)! for k ≥ 6. We can see that Conjecture 11.2.2 is a bold conjecture. An interesting small case is if W(7, 7) ∼ 7920, which is much larger than the known lower bound for W(7, 7). Note 2 that in Conjecture 11.2.1 Graham conjectures that for all k, W(k, k) ≤ 2 k . If it holds for 49 k = 7, then W(7, 7) ≤ 2 = 562,949,953,421,312. Now let us consider the red-blue coloring of [p] based on quadratic residues modulo p, where p is an odd prime. Similarly to Problem 4.4.1 in Chapter 4, we propose the following problem. Problem 11.2.3. Let g(k) be the largest odd prime p such that f(p) ≤ k − 1, where f(p) is the length of the longest AP in the quadratic residues modulo p. Does lim k→∞ g(k)/W(k, k) exist? If it exists, does it equal 0? The aim of this problem is to study whether we can obtain a good lower bound for W(k, k) by the red-blue coloring based on quadratic residues modulo p, where p is an odd prime. Maybe there is a constant c such that g(k) ≤ c k , and this is easier than proving W(k, k) ≤ C k for some constant C, but both are out of reach now. Maybe g(k) ≤ c1 (2 + ϵ)k for any ϵ > 0 and sufficiently large k. Rabung [227] constructed r-coloring using power residues and thereby gave some improved lower bounds on particular W r (k). He obtained lower bounds for some small van der Waerden numbers W(k, k), by computing all primes no greater than 20,117. By improving the efficiency of the algorithm of Rabung, Meilian Liang, Xiaodong Xu, Zehui Shao and Baoxin Xiu [183] performed the computation for all primes up to 6 × 107 , and obtained lower bounds on W(k, k) for k between 11 and 23. Daniel Monroe did more computations in a paper titled “New lower bounds for van der Waerden numbers using distributed computing”, in which he conjectured that lim

k→∞

W r (k) =r. W r (k − 1)

11.3 Set-coloring generalization of van der Waerden numbers |

115

In fact, it is far from our reach to prove that the limit in this conjecture exists and is finite. If we can prove this conjecture, then it is not difficult to see that the upper bound for W r (k) can be improved largely to cr k , and Conjecture 11.2.1 proposed by Graham holds for sufficiently large k. If we can prove that limk→∞ W r (k)/W r (k − 1) exists, and suppose this limit to be a, then, similarly, we can prove that W r (k) ≤ ca k . We can see this is much smaller than the upper bound for W(k, k) conjectured by Hendrick and Mehta.

11.3 Set-coloring generalization of van der Waerden numbers Before we propose new problems, we need the definition of set-coloring van der Waerden numbers. Let m > r ≥ 1 and k 1 , . . . , k m be integers, and C = {color i | 1 ≤ i ≤ m} be a color set. Let set-coloring van der Waerden number W (r) (k 1 , . . . , k m ) be the smallest positive integer n, such that if any integer in [n] is colored with an r-subset of C, then there must exist a k i -AP in which any integer is colored with an r-subset of C containing color i. Such a set-coloring generalization of van der Waerden numbers is proposed and studied in [299]. By studying such a set-coloring generalization, van der Waerden numbers may be understood better. On the other hand, computing setcoloring van der Waerden numbers can be regarded as a new challenge. In Chapter 16, we will discuss the set-coloring generalization of Ramsey numbers and Folkman numbers. The following Problem 11.3.1 does not seem easy to answer. If we can prove that the correct answer is yes, then the upper bound for W(k, k) based on the density method, (t) N(k, 1/2), cannot be the exact value of W(k, k). Note that W2t (k) ≤ N(k, 1/2) for any integers k ≥ 3 and t ≥ 1. (2)

Problem 11.3.1. W(k, k) < W4 (k) for any integer k > 2? (2)

(2)

In [299], it was obtained that W4 (3) = 10, and W4 (4) = 38. Note that W(3, 3) = 9, (4) N(3, 1/2) = 17, W(4, 4) = 35 and N(4, 1/2) = 57. We know that 38 ≤ W8 (4) ≤ 57, (4) and it is interesting to determine the exact value of W8 (4) or improve the known bounds. We can see N(3, 1/2) = 17 based on the data in [290] and can see N(4, 1/2) = 57 based on the data in [255]. We know that W(5, 5) = 178 and do not know the exact (2) value of N(5, 1/2). It does seem easy to determine the exact value of W4 (5). Let us propose a more general problem as follows. Problem 11.3.2. Is there a positive integer t such that (t)

W2t (k) ≤ N(k, 1/2) − a for some positive constant a and k sufficiently large? It is obvious that an affirmative answer to Problem 11.3.1 implies an affirmative answer (2) to Problem 11.2.2, because W4 (k) ≤ N(k, 1/2). We hope that the study on set-coloring

116 | 11 On van der Waerden numbers and Szemerédi’s theorem

van der Waerden numbers can help us to understand van der Waerden numbers better and improve the best known upper bound for W(k, k). (2) Computing the exact value of W3 (k) for small k is not easy. In [299], the exact (2) (2) (2) (2) values include W3 (4) = 13, W3 (5) = 25, and W3 (6) = 51 were obtained; W3 (7) ≥ (2) 119 and W3 (8) ≥ 160 were also obtained by computing. By a method of Rabung based on cubic residues of odd primes of the form p ≡ 1 (mod 6), more lower bounds (2) (2) (2) were obtained, including W3 (9) ≥ 777, W3 (11) ≥ 10,871, W3 (13) ≥ 18,805, (2) (2) (2) W3 (14) ≥ 33,164, and W3 (15) ≥ 58,815; W3 (8) ≥ 160 may be weak, and we suggest that the interested reader improve it.

11.4 The difference between various van der Waerden numbers It is not difficult to see that we can propose many problems and conjectures on van der Waerden numbers similarly to those on classical Ramsey numbers. The following four can be regarded as proposed this way. Problem 11.4.1. For any given positive integer s, lim

k→∞

W(k, k + 1) − W(k, k) =∞? ks

Conjecture 11.4.1. For given integer s ≥ 3, lim (W(s, k + 1) − W(s, k)) = ∞ .

k→∞

Now, we do not know if the limits in Problem 11.4.1 and Conjecture 11.4.1 exist and we cannot prove that there is a positive integer k 0 such that lim (W(s, k + k 0 ) − W(s, k)) = ∞ .

k→∞

Problem 11.4.2. For any given integer s ≥ 3, is W(s, k +1)− W(s, k) monotone-increasing with k? We cannot answer this question even for s = 3 and k sufficiently large. Note that c(

2 k 2 ) ≤ W(3, k) ≤ k dk . log k

The lower bound was proved by Yusheng Li and Jinlong Shu in [181], and the upper bound was proved by Bourgain in [49]. By the lower bound c(k/(log k))2 we can see that W(3, k) lim =∞. k k→∞ If the answer to Problem 11.4.2 is yes, then it is not difficult to see that lim W(s, k + 1) − W(s, k) = ∞

i→∞

11.5 Upper and lower bounds for r p (p 2 )

| 117

for any given s ≥ 3, i.e., Conjecture 11.4.1 holds. In fact, if W(s, 4) − W(s, 3) ≤ ⋅ ⋅ ⋅ ≤ W(s, k + 1) − W(s, k) , then

1 (W(s, k + 1) − W(s, 3)) . k−2 Therefore, if W(s, k + 1) − W(s, k) is monotone-increasing with k, then W(s, k + 1) − W(s, k) ≥

lim (W(s, k + 1) − W(s, k)) = ∞ .

k→∞

On the other hand, it seems that the following conjecture has not been proved yet. If the answer to Problem 11.4.1 is yes, then this conjecture holds. Conjecture 11.4.2. limk→∞ W(k, k + 1) − W(k, k) = ∞. We do not know how to prove that there is a positive integer k 0 such that lim W(k + k 0 , k + k 0 ) − W(k, k) = ∞ .

k→∞

It is not difficult to prove W(k + k 0 , k + k 0 ) − W(k, k) ≥ W(k 0 + 1, k 0 + 1) − 1 constructively. Similarly to Conjecture 6.4.1 in Chapter 6 on Ramsey graphs, we propose the following conjecture on van der Waerden numbers and related coloring. Conjecture 11.4.3. If k 1 , . . . , k t are integers and k 1 ≥ ⋅ ⋅ ⋅ ≥ k t ≥ 3, then for any positive integer n < W(k 1 , . . . , k t ), there is a (k 1 , . . . , k t )-coloring of [n] such that d1 ≥ ⋅ ⋅ ⋅ ≥ d t , where d i is the number of integers in color i in this (k 1 , . . . , k t )-coloring of [n]. It will be interesting even if Conjecture 11.4.3 could be proved only for n = W(k 1 , . . . , k t ) − 1 and a (k 1 , . . . , k t )-coloring of [n]. At the end of this section, we propose the following almost regular type conjecture related to van der Waerden numbers. Conjecture 11.4.4. For any positive integer n < W m (k), there is a good m-coloring C of [n] such that a i ∈ {⌊n/k⌋, ⌈n/k⌉}, where a i is the number of integers in color i in C.

11.5 Upper and lower bounds for r p (p2 ) In [287], Truss proved that n2 − p − ⌈(n − 1)2 /p⌉ ≤ r n (n2 ) < n2 − n − 12 √n + 2 , where p is the largest prime no greater than n. This upper bound in [287] may be weak, but we cannot improve it now. Truss was inclined to believe that the lower bound is

118 | 11 On van der Waerden numbers and Szemerédi’s theorem likely to be nearer to the exact value of r n (n2 ). The lower bound for r p (p2 ) in [287] is (p−1)2 . Brown and Freedman [52] proved that r p (p2 ) ≥ (p−1)2 +1 for any odd prime p. In [301], the following theorem on better lower bound for r p (p2 ) was proved. Theorem 11.5.1. For odd primes p ≥ 19, r p (p2 ) ≥ (p − 1)2 + t p , where limp→∞

tp (ln p)

= 1.

In general, we have r p (p s+1 ) ≥ (p − 1)r p (p s ), which was obtained by Szekeres no later than 1936 (see [112]). More similar results were proved by constructive methods in [301], as generalizations of the construction given by Szekeres. It seems an interesting challenge to improve this lower bound for r p (p s+1 ) further, and it may let us know more about the lower bound for r k (n), which is more general. Problem 11.5.1. For odd primes p and integer s ≥ 2, study the upper and lower bounds for r p (p s ). It does not seem easy to determine the order of r p (p2 )−(p −1)2 . It would be interesting to know the answer to the following problem. Problem 11.5.2. For odd prime p ≥ 3, r p (p2 ) − (p − 1)2 = O(p1/2 )? For cyclic groups, we cannot prove the following conjecture now. Even worse, it does not seem easy to prove it in small cases by computation. For instance, does it hold when p = 11? Conjecture 11.5.1. For any odd prime p, r p (ℤ p2 ) = (p − 1)2 . Now let us discuss the different remarks on [301] in the Journal of Number Theory. The lower bound for r p (p2 ) was studied in a section in [301], where p is an odd prime. The paper was submitted to two other journals successively and was rejected by both before it was submitted to the Journal of Number Theory. One reviewer directly presented a very low evaluation of the problem on r p (p2 ), because he or she believed that mathematicians mainly research the asymptotic bound on r k (n) when n → ∞, and the problem on r p (p2 ) is of very little value. We guess that this reviewer believed so for two reasons. The main reason is that this problem has been researched only by very few people, and it does not seem important not only today, but also does not seem to have the potential to become important. The other reason is that the methods used in [301] are elementary, and no so-called advanced tools were used. Of course, the second reason is only a guess on our part. If so, then how do we evaluate those important works that are elementary in the history of mathematics? It is an important problem if reviews by very few mathematicians should be used in evaluating a mathematical paper, and how we should remark on their comment if they are inappropriate. The part on Euler’s work on number theory in Weil’s masterpiece on the history of number theory titled “Number theory: An approach through history from Hammurapi to Legendre”, is full of admiration for the master. Note that Weil was one of the most greatest mathematicians in the twentieth century and was not a modest man. Then in such a book on the history of number theory, which he wrote in his old age, such

11.6 More problems |

119

admiration is even more unusual. Weil cited the remark without enthusiasm of Daniel Bernoulli on the work of Euler on number theory, and emphasized that Euler knew well how little interest his contemporaries, with the sole exception of Lagrange, had taken in his arithmetical work. The work on the number theory of Euler was not among the hotspots in his time. If a man of the same time as Euler considered the work of Euler on number theory according the same view as that reviewer of [301] mentioned above, he or she must regard this work as unimportant among Euler’s work. This is obviously not right for modern mathematicians. We believe that combinatorics is an important mathematical branch now (we find it more important than some mathematicians who believe that combinatorics is not important now), not only because of the work of combinatorists for the past few decades, or the relation between combinatorics and computers, but also because it is a potential counterpart of calculus. We are not saying that combinatorics will take off soon; after all, it is very difficult. However, sooner or later, it will develop into something more important. We hope it will not be too long, even though it does not matter if it takes a long time. Calculus has been developing for more than 300 years. We believe that, after the great development of combinatorial mathematics, the unity of mathematics will become more obvious with time, and it will not be necessary to preach such a unity as if it is a belief of little evidence.

11.6 More problems In this section, we will discuss more problems.

11.6.1 The Waerden–Szemerédi problem Similar to Ramsey–Turán theory, we may study the Waerden–Szemerédi problem, in which we consider how many elements a subset A of [n] can have if A is s-AP-free and [n] − A is t-AP-free. We may also study f(k, n) = min{max{g([n] − A(k, n))}}, where A(k, n) is a k-APfree subset of [n], max{g([n] − A(k, n))} is to find the length of longest AP in [n] − A(k, n), and min is to find the smallest one among these lengths during A(k, n) running over all best constructions related to r k (n), that is, A(k, n) is always a k-AP-free subset of [n], and |A(k, n)| = r k (n). So, by this definition, we know that there is a (k, f(k, n) + 1)-red-blue coloring of [n] with the maximum number of integers in red. We conjecture that there is (k, t)-red-blue coloring of [n] with fewer integers in red, where t < f(k, n) + 1. It would be interesting to know if f(k, W(k, k)−1) ≥ k. Let us propose the following conjecture.

120 | 11 On van der Waerden numbers and Szemerédi’s theorem Conjecture 11.6.1. If f(k, n) = min{max(g([n] − A(k, n)))}, then lim (f(k, W(k, k) − 1) − k) = ∞ .

k→∞

Supposing n = 177 = W(5, 5)−1, we would like to know the exact value of f(5, n). If we know the exact value of r5 (177) and a 5-AP-free subset of order r5 (177) in [177], then we can obtain an upper bound for f(5, n). However, we know only r5 (n) for n ≤ 110 now, and computing the value of r5 (177) must be much more difficult than computing the value of r5 (110). What we can do without much difficulty is to check subsets A of [177] for which the number of integers contained equals to the best known lower bound for r5 (177) and find a construction among them for which the length of longest AP in [177] − A is minimum, which may be very large. We know that r5 (177) ≥ 2r5 (36) + 2r5 (35) and r5 (36) = 25, r5 (35) = 24. Hence, r5 (177) ≥ 98. By the known results we know only r5 (177) ≤ 112. Similarly to Conjecture 5.2.4 in Chapter 5, there is a conjecture on van der Waerden numbers proposed by Landman [176]. Conjecture 11.6.2. For any integer k ≥ 3, W(k, k) ≥ W(k + 1, k − 1) ≥ W(k + 2, k − 2) ≥ ⋅ ⋅ ⋅ ≥ W(2k − 2, 2) . It is easy to see that W(2k − 2, 2) = 2k − 2. For any integer k ≥ 3, supposing that A1 = {2k − 3, 2k − 2} and A2 = [4k − 6]− A1 , then we can see that W(3, 2k − 3) > 4k − 6. Since 4k − 6 > 2k − 2, W(2k − 3, 3) = W(3, 2k − 3) > W(2k − 2, 2) for any integer k ≥ 3. Similarly, we may also conjecture that for any integer k ≥ 3, W(k + 1, k) ≥ W(k + 2, k − 1) ≥ W(k + 3, k − 2) ≥ ⋅ ⋅ ⋅ ≥ W(2k − 1, 2) .

11.6.2 The Anti-Szemerédi problem We can find the following problem in [108] (also [145]). Problem 11.6.1. Does there exist a set X of positive integers such that for some ϵ > 0 the following two conditions hold simultaneously: 1. For every finite Y ⊆ X there exists a subset Z ⊆ X, |Z| ≥ ϵ|Y|, which does not contain a 3-AP; 2. Every finite partition of X contains a 3-AP in one of its classes. There seems no known result on this problem.

11.6.3 A problem of Graham concerning monochromatic k-AP In [74] we can find the following problem of Graham, which was proposed earlier in [143]. Note that in this section, W(k) denotes W(k, k). Define W ∗ (k) to be the size of

11.6 More problems |

121

the smallest set X ⊆ Z such that any 2-coloring of X always has a monochromatic k-AP. Then, W ∗ (k) ≤ W(k). Graham proposed the following problem on W ∗ (k) in [143]. Problem 11.6.2. Is W(k) − W ∗ (k) unbounded as k → ∞? Does lim

k→∞

W ∗ (k) =1? W(k)

Remark(s). W ∗ (3) ≤ W(3) = 9, W ∗ (4) ≤ 27, W(4) = 35. We can see that W ∗ (k) is similar to the size Ramsey number, for which we consider graphs with the arrowing property and the smallest edge numbers, but the graphs need not be complete. Letting B1 = {2, 3, 4, 5, 6} and B2 = {38 − i | i ∈ B1 }. Let A = {1, . . . , 37} − (B1 ⋃ B2 ). In [143], it was pointed out that A, a set of order 27, cannot be partitioned into two 4-AP-free subsets. By W ∗ (3) = W(3, 3), we may conjecture that for dense sets, the probability that such a phenomenon similar to W ∗ (4) < W(4, 4) happens will be larger. Hence, it is possible that Golomb rulers are different from W ∗ (4) for the property considered here. For W ∗ (k), we do not know whether this problem is very different in the cases k = 3 and k = 4, and to analyze the reason quantitatively is very interesting and seems difficult. It is not difficult to generalize W ∗ (k) to the multicolor case and the off-diagonal case. Let us consider another problem. We suggest doing a detailed analysis on W k (3) to study its lower bound. Due to its complexity, we regard this as between the computation of Sidon–Ramsey numbers and that of Schur numbers. We will discuss Schur numbers, Sidon–Ramsey numbers, and Golomb rulers in other chapters. We know that W5 (3) > 170 and W6 (3) > 223. An idea to improve these lower bounds is to search disjoint 3AP-free sequences. For W5 (3), if we can find 5 disjoint 3AP-free sequences of order 35 in [175], then we can improve the lower bound for W5 (3) to 176, and for W6 (3) we need to find six disjoint 3AP-free sequences of order 38 in [228]. One of our aims to study disjoint Golomb rulers is to find more methods to study disjoint k-AP-free sequences and Schur sets more quickly, and then obtain new lower bounds for the van der Waerden number W m (k) and the Schur number S(k). Of course, all of these seem difficult.

12 More problems of Ramsey type in additive number theory In this chapter, we will discuss more problems of Ramsey type in additive number theory. These problems include Hales–Jewett numbers, Schur numbers, and Rado numbers. The Sidon–Ramsey number is a special case of the Rado number, which will be discussed in the next chapter.

12.1 Problems related to the Hales–Jewett theorem The Hales–Jewett theorem was proved in [157] by A. W. Hales and R. I. Jewett. The Hales–Jewett theorem is believed to be one of the pillars of Ramsey theory, from which many other results follow, including the van der Waerden theorem. For every pair k, n of positive integers, let [k]n be the set of all sequences of length n having values in [k]. The elements of [k]n are referred to as words. Fix a letter x. A variable word is a finite sequence of length n having values in [k] ∪ {x}, where letter x appears at least once. If l is a variable word and i ∈ [k], then l(i) is the word obtained by substituting all appearances of the letter x in l by i. A combinatorial line of [k]n is a set of the form {l(i) | i ∈ [k]}, where l is a variable word. If A is a subset of [k]n , then its density is the quantity |A|/k n . The Hales–Jewett theorem asserts that for every r and every k there exists n such that every r-coloring of [k]n contains a combinatorial line. Let the Hales–Jewett number HJ(k, r) be the smallest n such that every r-coloring of [k]n contains a combinatorial line. The Hales–Jewett theorem is a generalization of van der Waerden’s theorem. As we know, van der Waerden’s theorem has a famous density version, conjectured by Erdős and Turán in 1936, proved by Szemeredi in 1975, and given a different proof by Furstenberg [128] in 1977. The Hales–Jewett theorem has a density version as well, proved by Furstenberg and Katznelson in 1991 by means of a significant extension of the ergodic techniques that were pioneered by Furstenberg in his proof of Szemeredi’s theorem. [224] titled “A new proof of the density Hales–Jewett theorem” by Polymath and [223] titled “Density Hales–Jewett and Moser numbers” are two interesting references on the density Hales–Jewett theorem. Note that the density Hales–Jewett theorem implies Szemerédi’s theorem. Polymath is a new joint research form that has been used in recent years. The interested reader can find much more related information without difficulty. In the polymath paper [224], the first elementary proof of the theorem of Furstenberg and Katznelson was given, together with a quantitative bound on how large n needs to be. In particular, it was shown that a subset of {1, 2, 3} n of density δ contains a combina-

https://doi.org/10.1515/9783110576702-012

124 | 12 More problems of Ramsey type in additive number theory torial line if n is at least a tower of 2’s of height O(1/δ2 ). In [217] Dodos, Kanellopoulos, and Tyros gave a simplified version of the polymath proof in [224]. Problem 12.1.1. Study the upper and lower bounds for the Hales–Jewett number HJ(k, r). The gap between the known upper and lower bounds for the Hales–Jewett number is very large, even in small cases. The interested reader may find some references without difficulty. It would be very interesting if we could prove that the upper bound for HJ(k, r) obtained by the density Hales–Jewett theorem must be larger than its exact value. Note that we have considered a similar problem on the upper bound for W(k, k).

12.2 Schur numbers and Rodo numbers We cite [144] as an important reference for this section. We will cite the definitions of Schur numbers and Rodo numbers here and discuss them in detail in the following sections. For integer r ≥ 2, an equation f(x1 , . . . , x r ) = 0 (or f(x1 , . . . , x r )) is partition regular if for any partition of the non-negative numbers in ℕ into finitely many classes C j , j ∈ [k], some C j0 contains a non-trivial solution to the equation. (Non-trivial means that not all the variables are equal.) We often think of partitions as colorings and the solution in a single class as monochromatic. For a partition regular equation f(x1 , . . . , x r ) and a positive integer k ≥ 2, let the Rado number f(k) be the smallest positive integer n such that if [n] is partitioned into k disjoint subsets C j , j ∈ [k], then there must be a solution (a1 , . . . , a r ) such that a i ∈ C j0 for any i ∈ [r] and some j0 ∈ [k]. We can see that if f(x1 , . . . , x r ) is partition regular, then f(k) is finite for any positive integer k. As we know, the Rodo number is a generalization of the Schur number. A rather complete theory of partition regularity for (systems of) linear equations was developed by Rado [228]. Rado was a student of Schur’s. For example, x+y = z is partition regular, but x + y = 3z is not. In fact, a single homogenous equation over ℕ is partition regular if and only if it has a non-trivial solution in 0󸀠 s and 1󸀠 s, (i.e., not all 0). Non-commutative Rado theory was discussed in [27]. Because we are focusing on problems of graph theory type or number theory type rather than group theory type, we will not discuss non-commutative type problems in detail here. Readers may find more interesting problems in [27]. We will discuss Schur numbers in Section 12.3, and Rado numbers in Section 12.4. We will focus on the problem that if f(x, y, z) = x2 + y2 − z2 is partition regular in Section 12.4.

12.3 Schur numbers

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12.3 Schur numbers Schur numbers are defined based on x + y = z and are difficult to compute. Let f(x, y, z) = x + y − z and S(k) = f(k) − 1 throughout this section. Note that in some references the Schur number is defined as f(k), for instance, in [311]. I. Schur proved that S(k) is finite for every positive integer k in [249]. He proved that S(k) ≤ k!e. He also proved that S(k + 1) ≥ 3S(k) + 1, and we can also find a proof in [250]. We know the value of S(k) for positive integer k ≤ 4; S(5) ≥ 160 was proved by Exoo in [116]. The lower bounds S(6) ≥ 536 and S(7) ≥ 1680 were obtained in [126]. The best-known lower bounds for Ramsey numbers, R 5 (3) ≥ 162, R6 (3) ≥ 538 and R7 (3) ≥ 1682, were obtained based on these lower bounds for Schur numbers. For the lower bound on R8 (3), we have R8 (3) ≥ 3R7 (3) + R5 (3) − 4 ≥ 5204 based on the inequality of Fan Chung. It would be interesting to improve this lower bound for R8 (3) by a new lower bound for S(8). It may be interesting for students to study better lower bounds on S(7) and S(8). If lim k→∞ [S(k)]1/k = ∞, then limk→∞ [R k (3)]1/k = ∞ because S(k) ≤ R k (3) − 2. In [311], it was proved that R k (3) ≤ (e − (1/6))k! + 1 based on R4 (3) ≤ 62, and S(k) ≤ (e − (1/6))k! − 1 for any k ≥ 4 was proved by S(k) ≤ R k (3) − 2. These upper bounds are the best-known ones now. They may be far away from the exact values. Problem 12.3.1. Study the upper and lower bounds for the Schur number S(k). For small cases, we suggest obtaining an interesting upper bound for S(5). We know no upper bound better than R5 (3) − 2 ≤ 305. It is natural to conjecture that this upper bound is a weak one. Even a small improvement without too much computation may be interesting and does not seem easy. In [126], some conjectures were proposed, one of them is the following one. Conjecture 12.3.1. The largest integer for which there is a symmetric sum-free partition into 5 sets is 160; and perhaps S(5) = 160. Now, let us discuss the weak Schur number. If x and y in x + y = z are required to be distinct, then we call the new number, similarly to the Schur number, the weak Schur number, denoted by WS(k). The upper bound for WS(k) known is now ⌊k!ke⌋+1 proved in [46]. It is obvious that WS(k) ≥ S(k). It would be interesting to know whether WS(k) > S(k) holds for every k sufficiently large. In 1952, G. W. Walker claimed that WS(5) = 196 without proof. In [94], WS(5) ≥ 196 was proved by constructing a partition of [196] of the required type. It remains as an open problem to prove the equality WS(5) = 196. If it holds, then we obtain an upper bound 196 for the Schur number S(5). Note that we do not know if S(5) < 305. We know that WS(6) ≥ 582 was obtained in [93]. A set of integers is k-wise sum-free (or k-SF) if it can be partitioned into k sumfree sets. Let n be a positive integer. Denote by g(n, k) the size of a largest k-SF subset

126 | 12 More problems of Ramsey type in additive number theory of [n]. It is not difficult to evaluate g(n, 1). In [2], Abbott and Wang indicated how one may evaluate g(n, 2) for n < 54 and obtained some general upper and lower bounds for g(n, k). Problem 12.3.2. Study the upper and lower bounds for g(n, k). It is not difficult to see that g(n, k) may be used in computing bounds or values of small Schur numbers. We may also define numbers similar to g(n, k) for k-AP-free sets and use them in computing bounds for van der Waerden numbers. Similar to the Schur number S(k), the cyclic Schur number t(k) is defined to be the largest integer n, so that there is a partition of the integers in [n] into k classes, with no solution to the congruence x + y ≡ z (mod (n + 1)). We can see that S(k) ≥ t(k). In fact, Exoo proved S(5) ≥ 160 [116] by proving t(5) ≥ 160. Conjecture 12.3.2. S(k) = t(k). It is known that Conjecture 12.3.2 holds for positive integer k ≤ 4. On the other hand, it is not difficult to prove that t(k + 1) ≥ 2S(k) + 1. Therefore, if lim k→∞ [S(k)]1/k = ∞, then limk→∞ (t(k))1/k = ∞. In [2], H. L. Abbott and T. H. Wang proved that t(k + l) ≥ 2t(k)t(l) + t(k) + t(l) , for all positive integers k and l. It is interesting to study the lower bounds for Schur numbers, which may be a good way for studying the lower bound for R k (3), though this is also difficult. It would also be very interesting if we could obtain an upper bound for the Schur number S(k) given by smaller Schur numbers rather than by the multicolor Ramsey number R m (3). In particular, the following problem seems interesting. Problem 12.3.3. Does the inequality S(k) ≤ kS(k − 1) hold? Or does it hold for any k that is sufficiently large?

12.3.1 A problem on sum-free subset of a given set In article “Paul Erdős and the probabilistic method” [7], Alon mentioned the following problem. Problem 12.3.4. Does every set of n non-zero integers contain a sum-free subset of cardinality at least n/3 + w(n), where w(n) tends to infinity with n? This problem remains open, and Alon believes that it would be extremely surprising if there were no such w(n). In [92] it was proved that the constant 1/3 is tight. So, even if w(n) in Problem 12.3.4 does tend to infinity with n, limn→∞ w(n)/n = 0.

12.4 Rado numbers | 127

12.4 Rado numbers There are many unsolved problems on Rado numbers. Similar to the cases in other problems in Ramsey theory, the values of the Rado numbers are far from being reached in many cases, even for linear equations. Rado numbers for some linear equations were studied in a some papers. For example, in [241], for f(x, y, z, w) = x + y + kz − lw, the related two-color four-variable Rado numbers were determined for some positive integers k and l. Before we discuss the main problems in this section, let us cite a paragraph in the Abstract and a problem from [23]. In the Abstract of [23] some results were surveyed as cited below. A system of linear equations with integer coefficients is partition regular over a subset S of the reals if, whenever S−{0} is finitely colored, there is a solution to the system contained in one color class. It has been known for some time that there is an infinite system of linear equations that is partition regular over ℝ but not over ℚ, and it was recently shown (answering a long-standing open question) that one can also distinguish ℚ from ℤ in this way.

Now let us cite the following problem in [23]. Problem 12.4.1. If G and H are subgroups of ℚ such that G does not contain a subgroup isomorphic to H, must there exist a system (of linear equations with integer coefficients) that is partition regular over H but not over G? For nonlinear equations, as stated by Graham, the situation is much less clear than for linear equations. In [73], Croot proved a conjecture of Erdős and Graham on sums of unit fractions. It is a striking result. Let us discuss this theorem now.

12.4.1 A theorem of Croot on sums of unit fractions Let X and Y be two integers and X < Y. Denote [X, . . . , Y] by [X, Y]. Let us cite the theorem of Croot on sums of unit fractions. Theorem 12.4.1. There exists a constant b > 0 such that if we r-color the integers in [2, b r ], then there exists a monochromatic set S such that ∑n∈S 1/n = 1. Croot proved Theorem 12.4.1 as a corollary of the main theorem in [73]. Before we cite the main theorem, we need to introduce some notations and definitions first. Define C(X, Y; θ) to be the integers in [X, Y] all of whose prime power divisors are ≤ X θ and let C󸀠 (X, Y; θ) be those integers n ∈ C(X, Y; θ), such that ω(n) ∼ Ω(n) ∼ log log n , where ω(n) and Ω(n) denote the number of prime divisors and the number of prime power divisors of n, respectively. The following is the main theorem in [73].

128 | 12 More problems of Ramsey type in additive number theory Theorem 12.4.2. Suppose C ⊂ C󸀠 (N, N 1+δ ; θ), where θ, δ > 0, and θ + δ < 1/4. If N ≫θ,δ 1 and 1 ∑ >6, n n∈C then there exists a monochromatic set S ⊂ C for which ∑n∈S 1/n = 1. To prove Theorem 12.4.1 based on Theorem 12.4.2, Croot showed that for r sufficiently large, 1 ∑ > 6r , n 󸀠 1+δ n∈C (N,N

;1/4.32)

where N = e163,550r and N 1+δ = e166,562r . Thus, if we partition the integers in [2, e167,000r ] into r classes, then for r sufficiently large, one of the classes C satisfies the hypotheses of Theorem 12.4.2, and Theorem 12.4.1 follows. Croot pointed out that b may be taken to be much smaller. On the other hand, Croot also noted that b cannot be taken to be smaller than e, since the integers in [2, e r−o(1) ] can be placed into r classes in such a way that the sum of reciprocals in each class is just under 1. Maybe we can obtain a larger lower bound on b based on this idea by more detailed analysis. More generally, we propose the following problem. Problem 12.4.2. Study the value and bounds on the smallest b such that if we r-color the integers in [2, b r ], then there exists a monochromatic set S such that ∑n∈S 1/n = 1. We can see that the proof of Theorem 12.4.1 is not easy. It may be much more difficult to determine the value or obtain good upper and lower bounds on the smallest b in Problem 12.4.2.

12.4.2 On x 2 + y 2 = z2 One of the most famous problems on nonlinear Rado problems is the following problem of Erdős and Graham, which has been open for more than 30 years (see [144]). Problem 12.4.3. Is f(x, y, z) = x2 + y2 − z2 partition regular? Just as pointed out by Graham in [144], there is actually very little data (in either direction) to know which way to guess. If f(x, y, z) = x2 + y2 − z2 is partition regular, then f(k) is finite for any positive integer k, and we may propose the following problem. Problem 12.4.4. Suppose that f(x, y, z) = x2 + y2 − z2 is partition regular. Study the exact value and bounds for f(k). It is easy to see that f(1) = 5. The lower bounds for f(2) that are smaller than 7824 were obtained before [164]. In [164], it was proved that the set [7824] can be partitioned into two parts, such that no part contains a Pythagorean triple x2 + y2 = z2 , while this is impossible for [7825]. That is, f(2) = 7825.

12.5 Rado numbers in group theory

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Now let us propose the following problem. Problem 12.4.5. Compute the value of f(3). It is natural to conjecture that f(3) may be much larger than f(2). Problem 12.4.6. Study the upper and lower bounds for f(k + 1)/f(k), where f(x, y, z) = x 2 + y2 − z2 . We can construct a graph G in the following way. The vertex set is the positive integer set ℕ, and for any pair integers x and y in ℕ, we add an edge between them if and only if there is an integer z such that x2 + y2 = z2 or x2 + z2 = y2 . We can see that in such a graph G, for any given positive integer n0 , there is a vertex in G with degree no a a smaller than n0 . In fact, suppose that x = 2a0 q1 1 . . . q t t , and for any integer m such that m | (x/2), let n = x/(2m), then both (m, x) and (n, x) are edges in graph G. We may study the structure of K4 and K4 − e in graph G. Now let us consider the following problem. Problem 12.4.7. Suppose that f(x, y, z) = x2 + y2 − z2 . Study the lower bound for f(k) obtained by greedy algorithm. We cannot prove the following conjecture proposed by Erdős, Sós, and Sárközy in [109]. Conjecture 12.4.1. For any k ∈ N and any k-coloring of N, there are infinitely many monochromatic sums a + a󸀠 , which represent a square a + a󸀠 = x2 . It is not difficult to see that if f(x, y, z) = x2 + y2 + z2 is partition regular, then Conjecture 12.4.1 holds.

12.5 Rado numbers in group theory The following problems on r3 (G), an analog of r3 (n) in group theory, are cited from an unpublished article of Croot titled “Research problems in arithmetic combinatorics”. For a group G define r3 (G) to be the size of the largest subset of G containing no three-term arithmetic progressions. In this context, a three-term arithmetic progression is a solution to x + y = 2z, where x, y, z are distinct elements of G. For abelian groups G of odd order known upper and lower bounds for r3 (G) are appallingly far apart. However, perhaps there are non-abelian groups for which we can more easily deduce good upper and lower bounds. There are three problems related to this topic in the article of Croot. Problem 12.5.1. Does there exist an infinite family of non-abelian groups G of odd order for which r3 (G) > |G|/ log T |G|?

130 | 12 More problems of Ramsey type in additive number theory

Problem 12.5.2. Does there exist an infinite family of non-abelian groups G of odd order for which one can prove good upper and lower bounds for r3 (G) – bounds that differ by a constant factor, or even asymptotic bounds? The third problem is due to various people, implicit in a problem of Bourgain and Tao. A consequence of a theorem of Roth and Meshulam is that if G = ℤ3 × ⋅ ⋅ ⋅ × ℤ3 , (n copies of ℤ3 in all), then 3n r3 (G) ≪ . n Consider the following problem. Problem 12.5.3. Show that r3 (G) = o(3n /n).

13 Sidon–Ramsey numbers Disjoint Golomb rulers were systematically studied in [169]. Some results on disjoint Golomb rulers can be found in [169] and [260]. The Sidon–Ramsey number was defined in [185]. In general, it seems difficult to obtain good bounds on disjoint Golomb rulers without a deep theoretical analysis. We may try to improve the known bounds for disjoint Golomb rulers based on Sidon–Ramsey numbers. Of course, this seems rather difficult. On the other hand, disjoint Golomb rulers are useful in studying Sidon–Ramsey numbers. We will propose some conjectures and problems on Sidon– Ramsey numbers in this chapter, without discussing many details of disjoint Golomb rulers.

13.1 Basic definitions Let us cite the definitions of the Sidon set and the Golomb ruler first. Definition 13.1.1. A set {a i | 1 ≤ i ≤ k} of integers is a Sidon set if the sums a i + a j , for i ≤ j, are all different. Definition 13.1.2. A k-mark Golomb ruler is a set of k distinct non-negative integers, also called marks, {a i | 1 ≤ i ≤ k}, such that all differences a i − a j , i ≠ j are distinct. The difference between the maximal and minimal integers is referred to as the length of the Golomb ruler. The Golomb ruler was named after Solomon W. Golomb and was discovered independently by Sidon and Babcock. It is not difficult to see that Golomb rulers and Sidon sets are part of the same problem. Golomb rulers give various important applications in engineering. Sidon studied them as a mathematical problem. In the combinatorial number theory part of the Handbook of Combinatorics [226], the Sidon set is surveyed as an important topic. Definition 13.1.3. Let f(N) denote the maximum integer k for which there exists a Sidon set {a i | 1 ≤ i ≤ k} such that 1 ≤ a1 < ⋅ ⋅ ⋅ < a k ≤ N. The Sidon number F(k) = min{N | f(N) = k}. We list the values of the known Sidon numbers in Table 13.1. Computing the upper bound for F(k) is difficult for large k here. Definition 13.1.4. We say that I disjoint Golomb rulers (DGR) each being a J-subset of [n] is a (I, J, n)-DGR. Let H(I, J) be the smallest n such that there is an (I, J, n)-DGR. It is obvious that H(I, J) ≥ IJ. An (I, J, n)-DGR is regular if n = H(I, J) = IJ. We also say that H(I, J) is regular if H(I, J) = IJ. Computing the exact value of H(I, J) may be

https://doi.org/10.1515/9783110576702-013

132 | 13 Sidon–Ramsey numbers Tab. 13.1: Values of some Sidon numbers F(k), 2 ≤ k ≤ 27. k F(k)

2 2

3 4

4 7

5 12

6 18

7 26

8 35

9 45

10 56

11 73

12 86

13 107

14 128

k F(k)

15 152

16 178

17 200

18 217

19 247

20 284

21 334

22 357

23 373

24 426

25 481

26 493

27 554

difficult, even for I, J not large. In general, to determine whether H(I, J) = IJ is not easy either [260], if J is neither small nor much smaller than I. The Sidon–Ramsey number was defined as follows [185]. Definition 13.1.5. For any positive integer k, the Sidon–Ramsey number SR(k) is the smallest positive integer n such that in any k-coloring of [n], there is at least one monochromatic subset that is not a Sidon set. We can see that the Sidon–Ramsey number is a special case of the Rado number.

13.2 On small Sidon–Ramsey numbers We know that H(7, 9) = 63 and SR(7) > 63. We can obtain SR(7) ≥ 65 by the following construction, in which the first Golomb ruler is of order 10 and the others are of order 9. We choose the first ruler at random, and search six disjoint ones in [64] that disjoin with the first one. Then we obtain the following construction. 2, 13, 22, 28, 29, 47, 51, 59, 61, 64 . 16, 19, 20, 26, 35, 37, 49, 57, 62 ; 10, 11, 18, 21, 40, 42, 46, 55, 60 ; 6, 12, 15, 31, 36, 43, 44, 54, 58 ; 5, 9, 17, 23, 30, 32, 33, 52, 63 ; 3, 4, 8, 14, 27, 39, 41, 48, 56 ; 1, 7, 24, 25, 34, 38, 45, 50, 53 . We know that SR(7) ≤ 70. It does not seem easy to determine the exact value of SR(7) and SR(8) without much computation. Problem 13.2.1. Study the exact values and bounds for SR(7) and SR(8). At the end of this section, let us propose the following almost regular type conjecture related to Sidon–Ramsey numbers, which is similar to Conjecture 11.4.4 in Chapter 11 related to van der Waerden numbers. Conjecture 13.2.1. For any positive integer n < SR(k), there is a good k-coloring C of [n] such that a i ∈ {⌊n/k⌋, ⌈n/k⌉}, where a i is the number of integers in color i in C.

13.3 Upper bounds for Sidon–Ramsey numbers |

133

Let us consider an example. If Conjecture 13.2.1 holds, and SR(7) > 65, then there is a good construction {U i | 1 ≤ i ≤ 7}, such that ⋃7i=1 U i = [65], U i ∩ U j = 0 for different i, j ∈ [7], where U i is a 10-ruler for any i ∈ {1, 2} and U i is a 9-ruler for any i ∈ {3, 4, 5, 6, 7}. This can make the computation simpler.

13.3 Upper bounds for Sidon–Ramsey numbers {X1 , . . . , X k } is called a (k, n)-Sidon–Ramsey construction, if X1 , . . . , X k are disjoint Sidon sets, and ⋃ki=1 X i = [n]. A Sidon–Ramsey construction is abbreviated to SRC, and a (k, n)-Sidon–Ramsey construction is called a (k, n)-SRC. It is not difficult to prove the existence of SR(k) for any positive integer k based on the known upper bounds on Sidon numbers. In fact, we can give SR(k) an upper bound this way. Suppose integer k ≥ 2. Since F(t)/t tends to infinity as t tends to infinity, so for any given positive integer k, there is t0 such that (F(t0 ) − 1)/(t0 − 1) > k. Suppose that Sidon sets A1 , . . . , A k are k disjoint subsets of [F(t0 ) − 1]. Note that in [F(t0 ) − 1], there is no Sidon set of order t0 . Therefore, SR(k) ≤ F(t0 ) when (F(t0 ) − 1)/(t0 − 1) > k. Similarly, SR(k) ≤ (t0 − 1)k + 1 when (F(t0 ) − 1)/(t0 − 1) > k. In [185], the following result on Sidon–Ramsey numbers was proved. Theorem 13.3.1. For any integers k, t ≥ 2, SR(k) exists, and if (F(t) − 1)/(t − 1) > k, then SR(k) ≤ (t − 1)k + 1. Now let us give an upper bound for the Sidon–Ramsey number SR(k) for small k ≥ 6 based on this theorem. Note that we will use the results on the values of Sidon numbers given in Table 13.1. The value of SR(k) was computed in [185] for k ∈ {2, 3, 4, 5}. Let us consider some examples. We know that F(10) = 56, and (56 − 1)/(10 − 1) > 6, thus SR(6) ≤ 55 = (10 − 1) × 6 + 1. We can obtain upper bounds for more Sidon– Ramsey numbers similarly. Since H(5, 9) = 54 and F(10) = 56, it is not difficult to see that SR(6) ≤ 53. Since H(7, 10) ≥ 71 = H(6, 10) + 1 and F(11) = 73, we have SR(7) ≤ 70. On the other hand, if H(I, J) = IJ, then S(I) ≥ IJ +1. Therefore, we can obtain lower bounds for some Sidon–Ramsey numbers based on related known results on regular DGRs. For instance, in [169] it was proved that if p is an odd prime, then H(p + 1, p) = p2 + p. Therefore, we have S(p + 1) ≥ p2 + p + 1 when p is an odd prime. We will list these upper bounds together related lower bounds in Table 13.2. It seems that these bounds are weak. Most lower bounds for SR(k) in [185] seem weak. We obtain better lower bounds in some cases, which are included in Table 13.2. It seems difficult to obtain interesting results this way, for it is not easy to solve the related disjoint Golomb ruler problems. Let us consider the following problem.

134 | 13 Sidon–Ramsey numbers Tab. 13.2: Upper and lower bounds for SR(k), 6 ≤ k ≤ 19. k upper bound lower bound

6 55 50

7 70 65

8 97 81

9 118 97

10 141 114

11 166 133

12 193 157

k upper bound lower bound

13 235 170

14 267 186

15 301 190

16 337 209

17 355 229

18 415 306

19 457

Problem 13.3.1. Study the upper and lower bounds for the Sidon–Ramsey number SR(k). Maybe SR(k) > k 2 for every integer k ≥ 3, but it is far from being reached now. For instance, SR(14) > 196? In [185], the following problem was proposed. Problem 13.3.2. SR(k) ≥ k 2 + 2k for any integer k ≥ 2? It is known that SR(k) ≥ k 2 + 2k holds for every k such that 2 ≤ k ≤ 8. We can establish some constructive results for Sidon–Ramsey numbers without difficulty. For instance, we can see that SR(s + t) ≥ SR(s) + SR(t) − 1, which is often weak. We can prove the following constructive results. Theorem 13.3.2. If a and b are positive integers and a ≥ b, then (1) SR(a + b) ≥ SR(a) + SR(b) + b − 1; in particular, SR(2a − 1) ≥ SR(a) + SR(a − 1) + a − 1; (2) SR(2a, t) ≥ 2 SR(a, t − 1) + 2a. Proof. (1) Let F 0 = {X1 , . . . , X a } be an (a, n)-SRC, and F1 = {Y1 , . . . , Y b } be a (b, m)SRC, where n = SR(a)−1 and m = SR(b)−1. For any i ∈ {1, . . . , a}, let W i = {x+SR(b)− 1 | x ∈ X i } and F2 = {W1 , . . . , W a }. For any i ∈ {1, . . . , b}, let U i = Y i ⋃{SR(a)+SR(b)− 2 + i} and F3 = {U1 , . . . , U b }. Therefore F2 ⋃ F3 is an (a + b, SR(a) + SR(b) − 2 + b)-SRC, and SR(a+b) ≥ SR(a)+SR(b)+b−1. If b = a−1, then SR(2a−1) ≥ SR(a)+SR(a−1)+a−1. (2) can be proved similarly.

13.4 On Golomb rectangles Let us discuss Golomb rectangles in this section. We write this section on Golomb rectangles because in computing Golomb rectangles, the Drawer principle is often used to make the computation easier. A Golomb rectangle is an N × M array of ones and zeros such that the differences between the positions of every pair of ones in the rectangle, considered as vectors, are distinct. Let G(n, k) be the maximum number of ones that can be present in an N × M Golomb rectangle.

13.4 On Golomb rectangles |

135

In [259], Shearer proved the following lemma and pointed out that this means that searches for Golomb rectangles can be performed with a modified version of his Golomb ruler search program. Lemma 13.4.1. N × M Golomb rectangles with K ones correspond 1 − 1 with K element Golomb rulers with elements chosen from the set {i + (2N − 1)(j − 1) | 1 ≤ i ≤ N; 1 ≤ j ≤ M}. Let us consider an example. It is known that G(27, 6) ≥ 18 (see [254]). To prove G(26, 6) ≥ 18, we need only find an 18-ruler with elements chosen from the set {i + (2N − 1)(j − 1) | 1 ≤ i ≤ N; 1 ≤ j ≤ M}, where N = 26, M = 6. That is, we need to find an 18-ruler in the following set A: A = {1, . . . , 26} ⋃{52, . . . , 77} ⋃{103, . . . , 128} ⋃ {154, . . . , 179} ⋃{205, . . . , 230} ⋃{256, . . . , 281}. We have done some computation to search for a construction that guarantees G(26, 6) ≥ 18 but have not found it. We have to say that, if there is no construction that guarantees G(26, 6) ≥ 18, it seems difficult to prove it by computing. Let us propose the following general problem. Problem 13.4.1. Improve the known bounds for G(n, k).

14 Games in Ramsey theory It is natural to consider Ramsey-type games, for which the existence of the related function is often not difficult to know. For instance, Hex, also called Nash, can be regarded as a kind of Ramsey-type game. For more related discussions, readers can find [129] and some papers citing it. We can see that even for such a game as Hex, which seems simpler than those discussed in this chapter, it is difficult to give a general winning strategy, even for a small game. A survey on combinatorial games can be found in [125].

14.1 The Ramsey graph game The Ramsey graph game is played by two players on edges of the complete graph K N on N vertices. The two players alternately take turns at claiming some number of unclaimed edges until all edges are claimed. One of the players, called Maker, aims to create such a graph that possesses some fixed property P. The other player, called Breaker, tries to prevent Maker from achieving his goal; Breaker wins if, after all edges have been claimed, Maker’s graph does not possess P. A widely studied game of this kind is the q-clique game, where P = K q , and a graph possesses property P in the q-clique game if it contains a q-clique. An immediate question is how large q can be (in terms of N) such that Maker can achieve a K q in the game on K N . Some interesting problems and results can be found in some papers of Heidi Gebauer [130]. In [218], such a Maker–Breaker version of the Ramsey graph game, RG(n), was considered by Pekec, and a winning strategy for Maker requiring at most (n − 3)2n−1 + n + 1 moves was presented for N > (n − 2)2n−1 + 2. A competitive Ramsey graph game was also studied in [218]. In [218], Pekec proposed a conjecture on the Ramsey graph game, which states that if N < 2n−1 , then Breaker has a winning strategy for the RG(n) game. Pekec also mentioned that if this conjecture holds, then R(n, n) ≥ 2n−1 . This conjecture is not correct. The interested reader may read the original paper [218] for the conjecture and further discussion. Note that most conjectures on the bounds for R(n, n) have very little evidence, and it seems that this conjecture of Pekec is more or less an exception. Although it had been disproved, such a conjecture makes the Ramsey graph game more interesting for us. In [25], Beck proved the following theorem, by which we can see that the conjecture of Pekec cited in the previous paragraph cannot be true. Theorem 14.1.1. For some r = r(N) with r ∈ o(1), we have the following. If n ≤ ⌊2 log N − 2 log log N + 2 log e − 3 + r⌋, then Maker has a winning strategy in the RG(n) game. Otherwise, Breaker has a winning strategy.

https://doi.org/10.1515/9783110576702-014

138 | 14 Games in Ramsey theory

Theorem 14.1.1 is one of the main results in [25], of which the proof is long and difficult. There are some open problems on Ramsey games in [25]. Instead of citing any of them, we will propose a new problem. Although the conjecture of Pekec cited above is not correct, we still believe that it is an interesting idea to study the lower bound on R(n, n) through studying problems related to the RG(n) game. We propose the following Problem 14.1.1, mainly as an example of this idea. In any RG(n) game with partial coloring (E1 , E2 ), all edges in E1 are colored red and all edges in E2 are colored blue. Let E1 and E2 be two subsets of the edge set of G = K N , such that for any e1 ∈ E1 and e2 ∈ E2 , e1 and e2 share no common ends. Suppose that G[E1 ] is isomorphic to G[E2 ]. We can see that if N ≥ R(n, n), then in any RG(n) game with partial coloring (E1 , E2 ), Maker has a winning strategy. If we can prove that Breaker has a winning strategy for some large N0 , then we obtain a lower bound N0 for R(n, n) − 1. Problem 14.1.1. Let E1 and E2 be two subsets of the edge set of G = K N , such that for any e1 ∈ E1 and e2 ∈ E2 , e1 and e2 share no common ends. Suppose that G[E1 ] is isomorphic to G[E2 ]. Is there some N0 = n1+δ 2n/2 such that Breaker has a winning strategy in any RG(n) Game with partial coloring (E1 , E2 ) when N > N0 ? It is important to select good E1 and E2 . Even if the idea in Problem 14.1.1 works for some partial coloring (E1 , E2 ), the problem may be much more difficult than the problem related to Theorem 14.1.1.

14.2 Folkman games A maker–breaker version of the game based on Folkman numbers can be defined similarly to the Ramsey graph game, which will often be played on a graph that is not complete. Let us discuss only the vertex Folkman game, the FVG(n) game on a K n+1 free graph here; the edge Folkman game can be defined similarly. Suppose n is an integer and n ≥ 2. Let f(n) be the smallest positive integer N for which there is a K n+1 -free graph G of order N, such that Maker has a winning strategy in FVG(n) game on G. We can see that f(n) ≤ Fv (n, n; n + 1). It is obvious that f(n) ≥ 2n − 1, because if V(G) ≤ 2n − 2, then Maker can select no more than n − 1 vertices at the end of the game and cannot create a K n . Let G be the graph obtained by deleting a match of size n − 1 from K2n−1 . It is not difficult to see that Maker has a winning strategy in the FVG(n) game on G. Therefore, we have the following theorem. Theorem 14.2.1. For any integer n ≥ 3, f(n) = 2n − 1. If we want the FVG(n) game to be played on a regular graph, we can take the graph obtained by deleting a match of size n from K2n similarly.

14.3 The van der Waerden game | 139

Even we have determined the exact value of f(n), there are still other interesting problems on the FVG(n) game. We may also consider those cases in which the graph considered is not of the smallest order. For a non-trivial instance, we may ask if there is a regular K5 -free graph G of order Fv (4, 4; 5) − 1, such that in a FVG(4) game on G, Maker has a winning strategy. Note that we do not know the exact value of Fv (4, 4; 5) now. We may consider such a FVG(4) game on the unique (5, 3)-Ramsey graph. It is not obvious if Maker has a winning strategy on such a small graph, and we need do detailed analysis and computing. Let us consider the following problem. Suppose that n is a small positive integer. If G is a sparse K n+1 -free graph, on which Maker has a winning strategy in the FVG(n) game, then we may design an interesting FVG(n) game with G as a chessboard. The chessboard should be a graph with high symmetry. Of course, only when n and G are appropriate, is the game interesting. We may consider the case n = 5 first. Maybe it is more interesting if we suppose that G is a K n+1 − e-free graph in the FVG(n) game.

14.3 The van der Waerden game In [24], Beck defined and studied the following two-player game, the van der Waerden game, a game based on van der Waerden numbers. In the van der Waerden game, two players alternately pick previously unpicked integers in [N]. The first player wins if he or she has selected all numbers in a k-AP. Let W ∗ (k) be the smallest integer N, so that the first player has a winning strategy. In [24], it was proved that (W ∗ (k))1/k → 2 as k tends to ∞. Ramsey games on a uniform hypergraph were also studied in [24]. We may define a (k, N)-van der Waerden game in the following way. In a (k, N)van der Waerden game, two players A and B color the integers in [N] in turn, once an integer is to be colored. A uses red and B uses blue. If there is a k-AP in red, then A wins; if there is a k-AP in blue, then B wins. If there is no monochromatic k-AP when all integers in [N] are colored, then it is a draw. It is not difficult to see that W ∗ (k) is the smallest n such that it cannot be drawn at the end of the (k, N)-van der Waerden game. Problem 14.3.1. Study (k, N)-van der Waerden game and bounds for W ∗ (k). We can also study games of avoidance-type for the van der Waerden game. It is interesting to find out whether an avoidance-type van der Waerden game is useful in understanding the upper bound for W(k, k) better. Among the games discussed in this chapter, we would like to see the vertex Folkman game played by more people (they need not be mathematicians) in the future. We note that if we cannot understand the related topics in Ramsey theory better, it is not easy to understand the Ramsey-type games discussed in this chapter.

15 Local Ramsey theory In this chapter, we discuss local Ramsey numbers and define and study local Folkman numbers and local van der Waerden numbers. For local Ramsey numbers, we consider only the case of monochromatic complete subgraphs. Most known results on local Ramsey numbers are related to the local generalization of general Ramsey numbers. In most cases, it is not clear whether these topics will be important in improving the results on the related old topics in Ramsey theory because in those classical cases, we do not know whether or not local-type problems are equivalent to the old problems, and this needs more research.

15.1 Local Ramsey numbers A local k-coloring of a graph G is a coloring of the edges of G in such a way that the edges incident to each vertex v ∈ V(G) are colored with at most k different colors. The local Ramsey number R(G, s − loc) is defined as the smallest positive integer n such that any local s-coloring of K n contains a monochromatic copy of G. A (G, s − loc)Ramsey coloring is a local s-coloring of K n , where n = R(G, s − loc, t) − 1. We write R(K k , s − loc) as R(k, s − loc). In the following theorem proved in [154], the existence of the local Ramsey number R(G, s − loc) was proved, and an upper bound was obtained. Theorem 15.1.1. For every s ≥ 3 and m ≥ 2, R(K m , s − loc) ≤ ⌈ s

s(m−2)+1

s−1

⌉.

Similarly to R(G, s − loc), if we use no more than t colors in a local k-coloring, then we can define R(G, s − loc, t) as follows. Suppose that both s and t are positive integers and t ≥ s; R(G, s − loc, t) is defined as the smallest positive integer n such that any local s−coloring of K n by colors in {color 1, . . . , color t} contains a monochromatic copy of G. A (G, s − loc, t)-Ramsey coloring is a local s-coloring of K n with t colors, where n = R(G, s − loc, t) − 1. We write R(K k , s − loc, t) as R(k, s − loc, t). It is not difficult to see that the following inequalities hold. Theorem 15.1.2. If both s and t are positive integers and t ≥ s, then R s (G) ≤ R(G, s − loc, t) ≤ R(G, s − loc) . It is obvious that R s (G) = R(G, s−loc, s). It is interesting to study when the inequalities in Theorem 15.1.2 hold strictly. It is not difficult to know that R(G, s − loc, t0 ) = R(G, s − loc) for some t0 because R(G, s − loc) exists; R(G, s − loc, t) monotonically increases with t, and limt→∞ R(G, s − loc, t) = R(G, s − loc). In fact, because R(G, s − loc) exists, there is a (G, s − loc)-Ramsey coloring. Suppose in some (G, s − loc)-Ramsey coloring t0 colors are used. Then, R(G, s − loc, t0 ) = R(G, s − loc). On the other hand, we do not know if it holds for small t0 . https://doi.org/10.1515/9783110576702-015

142 | 15 Local Ramsey theory

Now let us prove the following theorem. Theorem 15.1.3. Suppose that s, t and k are positive integers no smaller than 3. Let R1 (k) = k. If t > s, then R t (k) − 1 ≥ (R(k, s − loc, t) − 1)(R t−s (k) − 1) . Proof. If G is a (k, s − loc, t)-Ramsey coloring, and H is a (k, t − s − loc, t − s)-Ramsey coloring, then it is not difficult to see that G[H] is a (k, t − loc, t)-coloring. Therefore, R t (k) − 1 ≥ (R(k, s − loc, t) − 1)(R t−s (k) − 1). Based on the inequality in Theorem 15.1.3, it is not difficult to prove the following result. Theorem 15.1.4. If s and k are positive integers and k ≥ 3, then lim (R s (k) − 1)1/s = lim (R(k, s − loc) − 1)1/s .

s→∞

s→∞

In [248], the following problem was proposed. This problem may be regarded as the most important problem on local Ramsey numbers. Problem 15.1.1. Does R k (m) = R(K m , k − loc) for all m ≥ 3 and k ≥ 2? The equality in Problem 15.1.1 was established in [154] for all m ≥ 3 when k = 2, but is not known for arbitrary m ≥ 3 when k ≥ 3 (see [248]). Similarly, we may ask when R k (m) < R(K m , k − loc, t) holds; it may never hold. Maybe we should consider the case k = 3 for m ≥ 3 first. This may be not a good way to answer if R k (m) = R(K m , k − loc) always holds, but it seems that we do not have any other feasible choice to understand the problem better now. In [154], it was shown that the well-known recursive upper bound R k (3) ≤ kR k−1 (3) − k + 2 is valid for R(K3 , k − loc) as well, and R(K3 , 3 − loc) = 17. Even for R4 (3) and R(K3 , 4 − loc, t), it seems there is nothing that is easy to compute, unless we can prove they are equal for some t = t0 . However, such an equality, if it holds, will be of little use in studying the related problem on multicolor classical Ramsey numbers and the related problem on Shannon capacity of graphs with bounded independence numbers, and we have to study the multicolor classical Ramsey numbers directly as before.

15.1.1 More inequalities on R(K m , k − loc, t) Suppose that s1 , s2 , t1 and t2 are integers no smaller than 3, and t1 ≤ t2 + s1 .

15.1 Local Ramsey numbers

| 143

If G is a (k, s1 − loc, t1 )-Ramsey coloring, and H is a (k, s2 − loc, t2 )-Ramsey coloring, then it is not difficult to see that G[H] is a (k, s1 + s2 − loc, t2 + s1 )-coloring. Therefore, we have R(k, s1 + s2 − loc, t2 + s1 ) − 1 ≥ (R(k, s1 − loc, t1 ) − 1)(R(k, s2 − loc, t2 ) − 1) . Then we have the following theorem, which is similar to Theorem 15.1.3. Theorem 15.1.5. If s1 , s2 , t1 and t2 are integers no smaller than 3, and t1 ≤ t2 + s1 , then R(k, s1 + s2 − loc, t2 + s1 ) − 1 ≥ (R(k, s1 − loc, t1 ) − 1)(R(k, s2 − loc, t2 ) − 1) .

15.1.2 Off-diagonal generalization of the local Ramsey number In the definition of the local Ramsey number only the diagonal case is considered. Now let us give R(K k , s − loc, t) an off-diagonal generalization. Suppose that both s and t are integers and t ≥ s ≥ 2. Let a1 ≤ ⋅ ⋅ ⋅ ≤ a s ; R loc (a1 , . . . , a s ; t) is defined as the smallest positive integer n such that for any local s−coloring of K n by colors in {color 1, . . . , color t}, and any vertex v, suppose that edges in {vu | vu ∈ E(K n )} are colored by colors in {color i1 , . . . , color i s }. Let the maximum order of the monochromatic complete subgraph containing v in color i j be w j , if we rearrange {w j | 1 ≤ j ≤ s} when it is necessary such that the {y j | 1 ≤ j ≤ s} obtained is monotone increasing, then y j ≥ a j for some j ∈ {1, . . . , s}. We can see that such a generalization includes R(G, s − loc, t) as a subcase, and its existence is easy to prove. In fact, we can see that R loc (a1 , . . . , a s ; t) ≤ R(K a s , s − loc, t) , when a1 ≤ ⋅ ⋅ ⋅ ≤ a s . It is not difficult to see that R loc (a1 , . . . , a s ; t) ≤ R loc (a1 −1, a2 , . . . , a s ; t)+⋅ ⋅ ⋅+R loc (a1 , . . . , a s−1 , a s −1; t)−s+2. We can give R(K k , s − loc, t) another off-diagonal generalization as follows. Suppose that s ≤ t−a. The mix-local Ramsey number R(G; H1 , . . . , H a ; s−loc, t) is the smallest n such that if we color K n with no more than t colors, where the subgraph induced by the edges in color i is H i -free for any i ∈ {1, . . . , a}, then the edges in other colors cannot form a (G, s − loc, t)-coloring. Similarly, we may generalize R(G; H1 , . . . , H a ; s − loc, t) to the more general form R(G1 , . . . , G b ; H1 , . . . , H a ; s − loc, t) based on the definition of R loc (G1 , . . . , G s ; t), which generalizes R loc (a1 , . . . , a s ; t). Let us consider R(K3 ; K3 ; s − loc, t) now. It is not difficult to prove the following theorem.

144 | 15 Local Ramsey theory Theorem 15.1.6. If s and t are integers, and 2 ≤ s < t, then 1 R t+c (3) − 1 ≥ (R(K3 ; K3 ; s − loc, t) − 1) ⌊ S(t + c − s)⌋ , 2 where S(t + c − s) is the t + c − s-th Schur number. Now, let us discuss the local generalizations of Folkman numbers and van der Waerden numbers in Section 15.2 and 15.3 respectively.

15.2 Local Folkman numbers Let us give Folkman numbers a local coloring type generalization and consider their existence. Let us consider the edge coloring case first. Suppose that both s and k are integers larger than 2; Fe (G, s − loc; K k ) is defined as the smallest positive integer n for which there is a K k -free graph H of order n, such that in any local s-coloring of H, there is a monochromatic copy of G. A complete proof of the existence of such a local edge Folkman number cannot be easy, which implies the existence of multicolor edge Folkman numbers. This is an exercise for the interested reader. We can define the local vertex Folkman number Fv (G, s − loc; K k ) similarly.

15.3 Local van der Waerden numbers Let us give van der Waerden numbers a local type generalization. For any i ∈ [n] and any j ∈ [n] − {i}, j is a k-neighbor of i in [n] if there is a k-AP in [n] that contains both i and j. Hence, in [n], j is a k-neighbor of i if and only if i is a k-neighbor of j. In a local (k, m)-coloring of [n], every integer in [n] is colored with one color in {color 1, . . . }, and for any i ∈ [n], the number of colors received by i and all its kneighbor is no more than m. Let the local van der Waerden number W(k, m − loc) be the smallest n such that any local (k, m)-coloring of [n] contains a monochromatic kAP. It is obvious that W(k, m − loc) ≥ W m (k). It is not difficult to prove the existence of W(k, m − loc). Let us give W(k, m − loc) an upper bound now. Theorem 15.3.1. W(k, m − loc) ≤ (k − 1)W m (k) + 1. Proof. Suppose that n = W(k, m − loc) − 1. Let d = [(n − 1)/(k − 1)]. Any pair of integers in [d+1], say a and b, are k-neighbors. In fact, if a < b, then a+(k−1)(b−a) ≤ a+(k−1)(d+1−a) = (k−1)d+1−(k−2)(a−1) ≤ n. Therefore, {a+(i−1)(b−a) | i ∈ [k]} is a k-AP in [n] because a + (k − 1)(b − a) ≤ n. If there is a local (k, m)-coloring of [n] without monochromatic k-APs, then W m (k) ≥ d+1 = [(n − 1)/(k − 1)]+1. So, W m (k) ≥ [(n − 1)/(k − 1)]+1 and n ≤ (k−1)W m (k). Hence, W(k, m−loc) ≤ (k−1)W m (k)+1. We propose the following problem on local van der Waerden numbers.

15.3 Local van der Waerden numbers |

145

Problem 15.3.1. Does W(k, m − loc) = W m (k) for all m ≥ 2 and k ≥ 3? It is interesting to consider the case in which m = 2 for all k ≥ 3. We may do some simple computation first to see if there are small cases in which the equality in Problem 15.3.1 does not hold. For instance, by Theorem 15.3.1 we know that W(3, 2 − loc) ≤ 2W(3, 3) + 1 = 19. We may study the exact value of W(3, 2 − loc) by computing. We leave it as an exercise for the reader.

16 Set-coloring Ramsey theory In most coloring problems in graph theory, we often color a vertex or an edge with one and only one color. Such a coloring was generalized by some graph theorists, such that every vertex or edge is not mapped to one color, but to a subset of the given color set [k]. In this chapter, we discuss the set-coloring generalization of Ramsey theory, in which a vertex or an edge is colored with a set of colors instead of with one color. We will discuss the set-coloring generalization of classical Ramsey numbers and vertex and edge Folkman numbers in this chapter. It is not difficult to generalize van der Waerden numbers (see [299]) and Folkman numbers (see [314]) and many other coloring problems to the set-coloring case. The set-coloring generalization of van der Waerden numbers is included in Chapter 11. There are many problems in Ramsey theory, some of which are generalizations of old ones. We believe that the set-coloring generalization of Ramsey theory discussed in this chapter is interesting. For instance, we may use some ideas on set-coloring vertex Folkman numbers to study Folkman type numbers with minimum chromatic numbers. The set-coloring generalization of the classical Ramsey number may be more interesting than most generalized Ramsey numbers.

16.1 Multigraph Ramsey numbers F. Harary and A. J. Schwenk studied Ramsey numbers for multigraphs in [160]. They concluded that it appears that there are no new interesting Ramsey numbers for multigraphs. They reached such a conclusion because they admitted edges between the same pair of vertices colored with same colors, and the multigraph generalization of the Ramsey number that they considered is the same as the related Ramsey number itself. Rather than coloring edges between the same pair of vertices with the same color, we need to add the non-degeneracy condition that edges between the same pair of vertices must be colored with different colors. (r) Let M n be the multigraph of order n, in which there are r edges between any two different vertices. Suppose that k and r are integers and k > r. In any k-edge-coloring (r) of M n , we always suppose that the edges between the same pair of vertices must be different colors. In [310], multigraph Ramsey numbers and their corresponding Ramsey graphs were defined and studied, in which the idea of set-coloring was used. Definition 16.1.1. Suppose q1 , q2 , . . . , q k and r are positive integers, where q i ≥ 2(1 ≤ i ≤ k), k > r. The multigraph Ramsey number f (r) (q1 , q2 , . . . , q k ) is defined to be the (r) minimum positive integer n such that in any k-edge-coloring of M n , there must be (r) i ∈ [k] such that M n has a complete subgraph of order q i , of which all the edges are in color i. https://doi.org/10.1515/9783110576702-016

148 | 16 Set-coloring Ramsey theory

Note that the set-coloring of edges in a complete graph is the same as the edgecoloring of the related multigraph, in which the edges between the same pair of vertices must be in different colors. So, f (r) (a1 , a2 , . . . , a k ) is not only the Ramsey number for multigraphs in which each pair of vertices are joined by r edges, but also the set-coloring Ramsey number. It is a generalization of the classic Ramsey number, because R(a1 , . . . , a k ) = f (1) (a1 , . . . , a k ). If k = r in Definition 16.1.1, then f (r) (q1 , . . . , q k ) = min{q1 , . . . , q k }. This is why we suppose k > r. It is easy to see that f (r) (2, q2 , . . . , q k ) = f (r) (q2 , . . . , q k ), so from now on we suppose q i ≥ 3 for any i ∈ [k]. It is obvious that f (r) (q1 , . . . , q k ) fits the commutative law, similar to Ramsey numbers. By f (r) (q1 , . . . , q k ) ≤ R(q1 , . . . , q k ) it is easy to know that f (r) (q1 , . . . , q k ) exists for any given integers q1 , . . . , q k and r < k. If q1 = ⋅ ⋅ ⋅ = q k = q, (r) we denote f (r) (q1 , q2 , . . . , q k ) as f k (q). Many methods and results on Ramsey numbers can be used in studying multigraph Ramsey numbers. Some graphs obtained in studying the lower bounds for Ramsey numbers can be used to give lower bounds for f (r) (q1 , q2 , . . . , q k ), although they may be weak. (r) In [310], a lower bound for f k (q) was proved by the probabilistic method. (r)

Theorem 16.1.1. If k > r ≥ 1, q ≥ 3, then f k (q) ≥ (k/r)(q−1)/2 (q!/k)1/q . (ri)

(r)

(2)

In [310], it was proved that f ki (q) ≥ f k (q), in particular, f4 (q) ≥ R(q, q). It is interesting to find out if

(2) f4 (q)

> R(q, q) holds for any integer q ≥ 3. (2)

Conjecture 16.1.1. If q is an integer and q ≥ 3, then f4 (q) > R(q, q). It is easy to prove this conjecture for q = 3. It is interesting to check this conjecture for more small cases. It may be not easy even for the case when q is 5 or 6.

16.2 Multigraph Ramsey numbers and other Ramsey-type problems The multigraph Ramsey number f (r) (q1 , q2 , . . . , q k ) may be regarded as a generalization of the topic in [9]. In [9], the following Ramsey-type parameter was considered. For each n and k, let m = f(n, k) denote the largest integer, so that for any k-coloring of the edges of K n , there exists a copy of K m whose edges receive at most k − 1 colors. It is not difficult to see that the problem above is nearly the same as the problem on (k−1) fk (m). In [310], the following theorem was proved. (k−1)

Theorem 16.2.1. For k ≥ 3 and q ≥ 3, if f k

(k−1)

(q) ≤ n < f k

(q + 1), then f(n, k) = q.

In [102], Erdős and Gyárfás proposed and studied the following problem. For fixed integers p, q, an edge-coloring of a complete graph K is called a (p, q)-coloring if the edges of each K p are colored with at least q distinct colors. Clearly, a coloring is a

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149

(p, 2)-coloring if and only if it is not an edge-coloring without monochromatic K p as subgraphs. Let f(n, p, q) be the minimum number of colors needed for a (p, q)coloring of K n . Note that we use (p, q)-coloring in this way only in this section. It is not difficult to see that the problem in [9] cited above is a special case of the problem in [102]. Erdős is an author of [9], who died in September 20th 1996, 5 days after [102] was submitted. [9] was published in 2000, and [102] was not cited in [9]. The other authors of [9] seemed not to have noticed that the problem in [9] cited above is a special case of the problem in [102], and they did not cite [102] in other papers before 2000. We would like to cite the following problem considered in [102]. Problem 16.2.1. Study the minimum positive integer n such that f(n, 4, 3) = 5. As pointed out in [102], it is not difficult to see that such a minimum integer n ≤ 17. (2)

16.3 Constructive lower bounds in Ramsey theory and f3 (q) (2)

We can obtain good lower bounds for multigraph Ramsey number f3 (q) by the probabilistic method, but we cannot construct a concrete construction for the classical Ramsey number R(q, q) of the same order. Note that R(q, q) may be much larger than (2) f3 (q). It is very interesting and it seems difficult to obtain some interesting progressions. Problem 16.3.1. Study the constructive lower bound on multigraph Ramsey num(2) ber f3 (q). An easier problem is to obtain a constructive lower bound for multigraph Ramsey numbers by the known methods used on classical Ramsey numbers. Maybe some young students would like to research these problems as exercises and the first step towards some difficult problems that are currently out of reach.

16.4 Set-coloring Folkman numbers Given a graph G and positive integers a1 , . . . , a k , we write G → (a1 , . . . , a k )vr (or G → (a1 , . . . , a k )er ) if for any k-coloring of V(G) (or E(G)) in which each vertex (edge) is colored with an r-subset of {1, . . . , k}, there exists a complete subgraph of order a i in which every vertex (or edge) is colored with an r-subset containing color i for some i ∈ {1, . . . , k}. For integer t > max{a1 , . . . , a k }, the set-coloring vertex and edge Folkman numbers are defined in [314] as follows. (r)

Fv (a1 , . . . , a k ; t) = min{|V(G)| | G → (a1 , . . . , a k )vr and K t ⊈ G} , (r)

Fe (a1 , . . . , a k ; t) = min{|V(G)| | G → (a1 , . . . , a k )er and K t ⊈ G} .

150 | 16 Set-coloring Ramsey theory (r)

(r)

We can define Fv (a1 , . . . , a k ; t) and Fe (a1 , . . . , a k ; t) without difficulty. If (r) (r) (r) a1 = ⋅ ⋅ ⋅ = a k = a, we denote Fv (a1 , . . . , a k ; t) (or Fe (a1 , . . . , a k ; t)) as Fv k (a; t) (or Fe Fe

(r) k (a; t)),

(r) k (a; t)).

(r)

(r)

and Fv (a1 , . . . , a k ; t) (resp. Fe (a1 , . . . , a k ; t)) as Fv

(r) k (a; t)

(or

It is not difficult to understand the existence of set-coloring vertex and edge Folkman numbers. In fact, we have the following two theorems on set-coloring vertex Folkman numbers, which are similar to those on multigraph Ramsey numbers in [310]. Theorem 16.4.1. If r ≥ 2 and t > max{a1 , . . . , a k+1 }, then (r)

(r)

(r−1)

Fv (a1 , . . . , a k ; t) ≤ Fv (a1 , . . . , a k , a k+1 ; t) ≤ Fv

(a1 , . . . , a k ; t) .

Theorem 16.4.2. If r, k, a, t, s are all positive integers, and t > a ≥ 2, then (r)

Fv k (a; t) ≤ Fv (1)

In particular, we have Fv k (a; t) ≤ Fv to the following problem.

(sr) sk (a; t)

(s) sk (a; t).

.

It is interesting to find out the solution

Problem 16.4.1. For which k, a and t, is there positive s0 such that for any s ≥ s0 , (s) Fv k (a; t) < Fv sk (a; t)? The most interesting case of this problem is, does (s)

Fv (a, a; a + 1) < Fv 2s (a; a + 1) hold for any integer a and s large enough? This seems difficult. Note that we cannot improve the best known upper bound for Fv (a, a; a + 1) obtained by the density method now. The off-diagonal generalization of Theorem 16.4.2 can be obtained similarly. For (2) instance, we have 13 = Fv (3, 4; 5) ≤ Fv (3, 3, 4, 4; 5). We may consider if G13 , the (2) (2) unique graph in Fv (3, 4; 5), is in Fv (3, 3, 4, 4; 5). If it is, then Fv (3, 3, 4, 4; 5) = 13, (2) otherwise Fv (3, 3, 4, 4; 5) > 13. The following theorem is similar to the result for r = 1 on vertex Folkman numbers. Theorem 16.4.3. If b 1 , b 2 , t1 , t2 are integers, b 1 ≤ t1 and b 2 ≤ t2 , then (r)

(r)

Fv (a1 , . . . , a k , b 1 + b 2 ; t1 + t2 + 1) ≤ Fv (a1 , . . . , a k , b 1 ; t1 + 1) (r)

+ Fv (a1 , . . . , a k , b 2 ; t2 + 1) . The following theorem is a generalization of the inequality Fv (a1 b 1 , . . . , a r b r ; ab + 1) ≤ Fv (a1 , . . . , a r ; a + 1)Fv (b 1 , . . . , b r ; b + 1) , cited in Chapter 8, and can be proved similarly. Theorem 16.4.4. If r ≥ 1, a1 , . . . , a k ≥ 2, b 1 , . . . , b k ≥ 2 and s ≥ max{a1 , . . . , a k }, t ≥ max{b 1 , . . . , b k }, then (r)

(r)

Fv (a1 b 1 , . . . , a k b k ; st + 1) ≤ Fv (a1 , . . . , a k ; s + 1)Fv (b 1 , . . . , b k ; t + 1) .

16.5 Some remarks

| 151

16.5 Some remarks We would like to cite some words of Weyl on generalizations written more than 60 years ago. In “A half-century mathematics” [298], Weyl pointed that “The real aim is simplicity: every natural generalization simplifies since it reduces the assumptions that have to be taken into account”. We can say that such a simplicity achieved by generalizations is an important way to unify mathematics when it becomes much larger. Let us explain the necessity of studying the set-coloring generalization of Ramsey theory. The best known general upper bounds on vertex Folkman numbers [89] and van der Waerden numbers [140] are given by the density method. Both of them improved the earlier results greatly. What can we say about the difference between these Ramsey-type numbers and the upper bounds obtained by the density method? To answer this question is an important aim to study the set-coloring generalization of Ramsey theory. We wish to prove that the difference between them is large, either by the set-coloring idea or other ideas, and find a method that is better than the density method to study the related upper bounds. We believe that these problems are of Ramsey type, and we lose the chance to understand them better in using the density method. Of course, we know that such an improvement will be very difficult. We believe that those related results obtained by the density method are an important improvement on these problems. When we consider constructive methods and probabilistic methods, we can say similar words. Set-coloring Folkman numbers were defined and studied in [314]. Some related Folkman-type numbers with minimum chromatic numbers were studied in [302]. Setcoloring van der Waerden numbers were studied in [299] and were discussed in Chapter 11 in this book. We believe that this research may be helpful in improving the known results of some related difficult problems. [310] received two reviews. We like to cite these reviews here because we believe that the reviewers are right. One of the reviews is as follows. “This article is very good, perfect and original results. So, I strongly recommend to publish on your journal.” The other review is as follows. “We believe the subject of this manuscript is quite interesting, and are surprised that nobody thought to generalize Ramsey numbers in this way before. (It seems that some work was done on bipartite multigraph Ramsey numbers, but according to the Authors’ references, this paper has not yet appeared.) We feel obliged to say that the results shown in this paper follow from standard techniques in elementary Ramsey theory, so the proofs here aren’t too surprising. From that point of view, we are reluctant to strongly recommend the paper for publication. However, this area could be of interest to many other researchers, and so we think the Authors have suggested an interesting problem for which they presented the beginning results. We therefore recommend that this manuscript be published as a note in Graphs and Combinatorics.” It is obvious that at that time, the reviewers did not know about the existence of [160], like the authors of [310]. A few years later, there are very few papers that

152 | 16 Set-coloring Ramsey theory

cite [310]. Although it might be cited a little more if it had been published in another so-called top journal, the difference would be very small. Even so, we believe that set-coloring generalization of Ramsey theory is interesting. Maybe it will become important in the future.

17 Other problems and conjectures In this miscellaneous chapter, we list some problems and conjectures in Ramsey theory that do not belong in any of the other chapters. What we cite in this chapter is far from complete, and the references may not be the newest. We include rainbows and anti-Ramsey type problems herein instead of in Chapter 10, because some related problems are on graphs and others are not.

17.1 Rainbows and anti-Ramsey-type problems Among anti-Ramsey-type problems, many results on rainbows have been proved. Let us cite [38] first. A coloring of the edge set of a graph G is called a b-bounded coloring if no color is used more than b times. We say that a subset of the edges of G is a rainbow if each edge is of a different color. A graph has property A(b, H) if every b-bounded coloring of its edges has a rainbow copy of H. In [38], the threshold for the random graph G n,p to have property A(b, H) was estimated. The anti-Szemerédi problem has been considered by some mathematicians. We discussed it in Section 11.6.1. There are more rainbow-type problems on arithmetic progressions, but we will not discuss them in detail. Rainbow-type results should not be considered as being completely anti-Ramsey type. In fact, they are also Ramsey type in essence. Fox, Grishpun, and Pach [123] showed that every 3-coloring of the complete graph on n vertices without a rainbow triangle contains a set of order Ω(n1/3 log2 n), which uses at most two colors, and this bound is tight up to the constant factor. They further showed that for fixed positive integers s, r with s ≤ r, every r-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Ω(n s(s−1)/r(r−1)(log n)c r,s ), which uses at most s colors [123]. It is interesting to determine the constant factors in these problems in [123].

17.2 Banach spaces and Ramsey theory Although none of the authors of this book is an expert on studying Banach spaces by methods in Ramsey theory, we would like to cite some related open problems and references in this section. We do so not because these works are famous, but because they are good examples that Ramsey theory can be a powerful tool in other mathematical branches, even in Banach spaces – a topic considered far from combinatorics before the work of Gowers and other mathematicians. Some related interesting problems were surveyed in [79].

https://doi.org/10.1515/9783110576702-017

154 | 17 Other problems and conjectures

As we know, a theorem is usually said to belong to Ramsey theory, in the words of Gowers, “if in any finite coloring of some mathematical object there is a sub-object of a certain type, which is monochromatic”. In Chapter 24 of the Handbook of the Geometry of Banach Spaces, Gowers surveyed Ramsey methods in Banach spaces. He points that there are, in fact, three (not completely distinct) ways in which Ramsey theory is useful in the study of Banach spaces. Let us cite them as follows. The first, and most obvious, is direct application of existing results from Ramsey theory. The second is the discovery of new results in Ramsey theory obtained with the specific aim of applying them to problems in the Banach space theory. The third way is the proof of pure Banach-space results that are Ramsey theoretic in character and inspired by the methods of Ramsey theory.

We believe that there are three similar ways in which Ramsey theory is useful in the study of other topics in mathematics. Instead of discussing the unsolved problems that Gowers discussed in his survey cited above, we will cite the following problem on the distance from a cube [79]. The Banach–Mazur distance between two isomorphic Banach spaces X and Y (not necessarily infinite-dimensional) is defined by dBM (X, Y) = inf {‖T‖ ⋅ ‖T −1 ‖ : T : X → Y is an isomorphism} . For every n ∈ ℕ define n n = max {dBM (X, l∞ ) : dim(X) = n} . R∞

Now let us consider a problem of A. Pełcayński [219]. n as n tends to infinity. Problem 17.2.1. Determine the asymptotic behavior of R∞ n are The best-known lower and upper bounds for R∞ n ≤ (2n)5/6 , c√n log n ≤ R∞

where c is an absolute constant. The lower bound is due to S. Szarek [282]. The upper bound is due to Youssef [315], which slightly improved the upper bound Cn5/6 due to A. Giannopoulos [136], where C is an absolute constant. n is a very challenging one, The problem of finding good upper estimates for R∞ see [48, 283]. This problem appears in the list in [79] because all proofs establishing a n use some variant of the Sauer–Shelah lemma, a comnon-trivial upper bound for R∞ binatorial result of Ramsey type.

17.3 The Erdős–Szekeres theorem | 155

17.3 The Erdős–Szekeres theorem Let us consider some problems of Ramsey type in combinatorial geometry in this section. We focus on the Erdős–Szekeres theorem and related problems. Morris and Soltan wrote a nice survey [202] on this topic in 2000. The Erdős–Szekeres theorem was proved in 1935 in [110] and is one of the most important papers in the history of Ramsey theory. This theorem is a solution to a problem proposed by Esther Klein, who became Szekeres’ wife in 1936. Esther Klein observed that any set of five points in a general position (i.e., no three of the points are on a line) in the plane contains four points that are the vertices of a convex quadrilateral. She suggested the following more general problem, namely the problem on the existence of a finite number N(n) such that from any set containing at least N(n) points in general position in the plane, it is possible to select n points forming a convex polygon (see [202]). We cite the Erdős–Szekeres theorem in the following Theorem 17.3.1. Theorem 17.3.1. Let n be a positive integer. Then there exists a least integer N(n) with the following property: If X is a set of N(n) points in the plane in general position (i.e., no three of which are collinear), then X contains an n-tuple, which forms the vertices of a convex n-gon. On the basis of the known value of N(n) for n ∈ {3, 4, 5}, Erdős and Szekeres proposed the following conjecture in their original paper [110]. Conjecture 17.3.1. N(n) = 1 + 2n−2 for all n ≥ 3. In [284], Conjecture 17.3.1 was proved for n = 6. In [111], Erdős and Szekeres proved that N(n) ≥ 1 + 2n−2 by constructing explicit examples. A better known upper bound n−2 is when n ≥ 7 is N(n) ≤ C2n−5 + 1 = O((4n )/(n1/2 )) (see [286]). In [288], Vlachos showed that n−2 n−3 n−3 N(n) ≤ C2n−5 − C2n−8 + C2n−10 +2.

Vlachos’ manuscript [288] led to further improvements in the following two papers. n−2 In [212], it was proved that lim supn→∞ N(n)/(C2n−5 ) ≤ 7/8, without using induction. Note that most upper bounds on N(n) were proved using induction. During the refereeing process of [288], each of the two authors of [199] independently fine-tuned the n−3 original arguments of [288] to get rid of the term C2n−10 . Their main result in [199] is the following, which is the best known general upper bound. Theorem 17.3.2. If n is an integer and n ≥ 6, then n−2 n−3 N(n) ≤ C2n−5 − C2n−8 +2.

In 2016, Andrew Suk [280] nearly settled Conjecture 17.3.1 by showing that N(n) ≤ 2/3 2n+6n log n for n ≥ n0 , where n0 is a large absolute constant. In the concluding remark

156 | 17 Other problems and conjectures

of [280], Suk states that Gábor Tardos improved the lower-order term in the exponent, showing that N(n) = 2n+O(√n log n) . There is also the question of whether any sufficiently large set of points in general position has an “empty” convex quadrilateral, pentagon, etc., that is, one that contains no other input point. In 1978 Erdős proposed this problem. For any integer n ≥ 3, determine the smallest positive integer H(n), if it exists, such that any set X of at least H(n) points in general position in the plane contains n points that are the vertices of an empty convex polygon, i.e., a polygon whose interior does not contain any point in X. It was proven by Horton in [165] that in the plane there are arbitrary large sets that do not contain empty convex 7-gons. This statement is due to an analytic construction of a planar set S k of 2k (k ≥ 1) points in general position determining no empty convex 7-gon. The interesting construction of Horton is simple. Nicolás [210] and Gerken [133] independently solved the problem for an empty convex 6-gon. Hence the empty convex n-gon problem has been completed. What was proved in [133] is that every set that contains the vertex set of a convex 9-gon also contains an empty convex hexagon. On the other hand, we may consider the k-interior point variant of the Erdős– Szekeres problem as follows. Conjecture 17.3.2. For any positive integer k, every point set in the plane with a sufficient number of interior points contains a convex polygon containing exactly k-interior points. This has been proved only for k ≤ 3. Let the minimum sufficient number in this conjecture be g(k). It is unknown whether g(k) exists for k ≥ 4. The best-known lower bound is g(k) ≥ 3k for k ≥ 3 obtained in [294], if g(k) exists. In [22] the line version of the Erdős–Szekeres theorem, that is, to find n lines in convex position in a sufficiently large set of lines that are in general position, was studied. Almost matching upper and lower bounds for the minimum size of the set of lines in general position that always contains n in convex position were obtained. This is quite unexpected, since as we saw above in the case of points, the best-known bounds are far from each other. It is interesting to know if we can do better for the line version of the Erdős–Szekeres theorem. Problem 17.3.1. Study the line version of the Erdős–Szekeres theorem to obtain matching upper and lower bounds for the minimum size of the set of lines in general position that always contains n in convex position. It would be interesting to know whether we can design an interesting game based on the Erdős–Szekeres theorem. For instance, if players A and B select point in R2 in turn, and supposing that they select a1 , b 1 , a2 , . . . , and so on. Let the player select the last point in some n-gon among a1 , b 1 , . . . be the loser of the game, and the other player

17.4 The chromatic number of the Euclidean plane |

157

be the winner. Can the players have a winning strategy such that the game will be over in fewer than 2n−2 + 1 turns? The Erdős–Szekeres problem gives us the impression that the methods used do not seem to work on other problem, unless the problem is similar to the Erdős–Szekeres problem. Until now, there is no trace of the methods having wide applications. Of course, this is not strange in combinatorics. On the other hand, if there had not been such a problem, then the history of Ramsey theory would have been different.

17.4 The chromatic number of the Euclidean plane This section is included in this chapter instead of Chapter 10, because it deals with infinite graphs instead of finite graphs, and it is related to set theory. All this makes it very different from the problems in Chapter 10. The most widely-known problem in Euclidean Ramsey theory is the chromatic number of the Euclidean plane 𝔼2 , which seems very difficult and relates to set theory. A survey of this problem can be found in [148]. Problem 17.4.1. Determine the value of χ(𝔼2 ), the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color. This question is known as the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson (it is attributed to Edward Nelson in [267, 268] and [269]). It is known that 4 ≤ χ(𝔼2 ) ≤ 7. Neither bound is difficult to obtain (see [148]) and they have been known since the time the problem was suggested. The correct value may actually depend on the choice of axioms for set theory (see [262] and [270]). In 1981, Falconer showed that if we assume the axiom that all subsets of 𝔼n are Lebesgue measurable, then χ(𝔼2 ) ≥ 5. Of course, this can tell us very little about the case in which some subsets in E n are not Lebesgue measurable. For other conditional results about χ(E2 ), see [270]. If we operate in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), then it follows by compactness that if χ(𝔼2 ) = r, then, in fact, there is a finite set that also requires r colors to legally color it. Many people may regard this as true without considering further in detail. However, as pointed out in [263], if we replace the axiom of choice by several nearly equally consistent axioms (dependent choice and every set is Lebesgue measurable), then we no longer have compactness, and the answer may change. It seems interesting to consider whether we can prove new results by taking sets that are not Lebesgue measurable as a focus. Maybe we should not suppose that every set is Lebesgue measurable. In [80] and [81], O’Donnell showed that for every integer g, there is a unit distance graph in 𝔼2 with girth greater than g, which has chromatic number 4. It is not easy to tell if this is evidence that χ(𝔼2 ) ≥ 5.

158 | 17 Other problems and conjectures There has also been some work on the bounds for χ(𝔼k ) and χ(ℚk ), where ℚ is the set of rational numbers. We will not discuss this in detail, and the reader may find related references without difficulty. We hope that these problems can lead us to more methods that can be used in studying other problems. For instance, problems in algebraic number theory.

Epilogue My elementary school days were spent at Beijing National Day School. After that I went to Beijing No. 66 Middle School and Bayi School. My journey with mathematics would start with Beijing No. 101 High School in 1962. At that time, I read Luogeng Hua’s Mathematical Induction and Discussion from Chongzhi Zu’s Evaluation the Ratio of the Circumference. I attended Hua’s lecture on “The mathematical problems related to honeycomb construction,” which had been reorganized and edited by Jiyi Yan, the teacher from the University of Science and Technology of China (USTC). (The names that I will mention later are either professors or students from USTC.) Because of my great appreciation of these papers, I applied to USTC in 1965, when Luogen Hua was Vice President and Director of the Department of Mathematics at USTC. When I enrolled to USTC, I was a little bit disappointed that Professor Hua was not teaching any class at the time. However, I interacted with four specific teachers at USTC. They were Sheng Gong, Jihuai Shi, Gengzhe Chang, and Zongxi Cai. Their work consisted of the Discussion on Hui Liu’s Cyclotomic Method, Average, Plural Geometry and The Isoperimetric Problem. My satisfaction was met. During the cultural revolution, I was assigned to Guizhou in 1970 after my graduation and relocated to Guangxi in 1972. Finally, I went to Guangxi Academy of Sciences in 1978. In 1979, with the connection to Weifan Hua, I went back to USTC with my enrollment in the advanced computer software program. At that time, I lived at Xing Zhang’s house. During this period, Zhengyou Zhang from the modern mechanic’s program was my classmate. I was most serious in Shuling Sun’s “Discrete mathematics” class. Yunqiu Shen taught me about “The bandwidth problem on graphs” class. After I came back to Guangxi, Yuanyuan Wang collaborated with me to translate the Theory and Problems of Discrete Mathematics. This was supposed to be the first published book about discrete mathematics in China. Then, Kenchen Zeng told me that Jiehua Mai had an excellent level of mathematics, so I wrote the paper about Several principles of the bandwidth problem on a graph with Mai and sent it to Dequan Chen for publication in a famous science journal called Acta Mathematics Sinica. In 1981, when Qiao Li furthered his studies in the United states, he wrote to me that Ramsey theory was a hot topic among American doctoral theses, which brought it to my immediate attention. After a period of enhancing my mathematical knowledge, I started to solve the Ramsey problem with Wenlong Su from Guangxi Wuzhou and Zhengyou Zhang and Kang Wu from South China Normal University. Zhenchong Li and Jiandong He from Guangxi Academy of Sciences also took part in this research program. Breakthroughs in 1996 revealed themselves with the first batch of important research results on Ramsey’s theorem, which were led by the group of Wenlong Su. This important compilation of results was finalized in a short paper by Qiao Li and published in the Chinese Science Bulletin. With the successful results of diagonal sehttps://doi.org/10.1515/9783110576702-018

160 | Epilogue

ries, Li wrote a paper and published it in Science China. Jingen Yang was responsible for editing and translating it – connected with overseas publishers who were specialized in editing and finalizing research papers. This esteemed group of academics included the following: Yunqiu Shen and Ronald L. Graham, the former President of the American Mathematical Society, who recommended the paper; it was published in Applied Mathematics Letters. Keqin Feng, Yinong Dong, Guozhong Dai, Lianhua Xiao, Shangzhi Li, Xianke Zhang, Yuyu Feng, Jiongsheng Li, Hongwen Lu, Zhishu You, Jianguo Zha – provided genuine assistance. Xiaodong Xu, a graduate student of the National University of Defense Technology, was an important component that became the main force of this scientific research and deserves special credit. With the accessibility of multitudes of resulting research with a continued effort to preserve the integrity of the research by rewriting the international best records for Ramsey’s theorem studies, and publishing many research papers in Discrete Mathematics, Discrete Applied Mathematics, Journal of Graph Theory, we protected the sustainability and longevity of the research. In retrospect, in 2002, we won the Guangxi Science and Technology Progress 1st prize and in 2006 we won Guangxi Science and Technology Progress 2nd prize. We were also awarded the Guangxi Science and Technology Achievement Award and Guangxi Computer Outstanding Achievement 1st prize. After graduating from USTC, I did not feel that it was advantageous to the higher end of things. Its teleological precedence was offset with my doubts of it being a passionate career. However, after 1978, I realized that USTC served as the anchor to my successes in mathematical research. It seems like I got help from USTC, from my professors and my old classmates. China’s present universities seem to fall into the Great Leap Forward that occurred in 1958. The whole education cast a blind eye to the expansion or unrealistic comparisons that caused a huge amount of long-term debts. The government’s official standard and money were the only priorities and parameters that mattered in the academic world. The fundamental purpose of running a school was lost. However, USTC is still heading down the right path, the university keeps its promise through generous means to stipulate and stimulate education and scientific research. I am very pleased and proud of the USTC’s progress in these endeavors. Currently, Xiaodong Xu is in charge of the major research along with the addition of Meilian Liang to our team. It has become increasingly difficult to improve the lower bounds for the classical Ramsey numbers with cyclic graphs, and further studies will be extremely difficult. A few discussions prevailed with Xiaodong Xu regarding the status of our work and its contents. The decisions among the following variables were discussed: (1) whether we should study lower bounds for classical Ramsey numbers by improving the old technique as before; (2) whether we should concentrate on using Cayley graphs to study the lower bounds for Ramsey numbers; or (3) the upper bound on the vertex Folkman number should be the new research focus. Xu proposed that from now on all we need to focus on is our interest and projects, and therefore, we shall focus on the new conjecture and the corresponding study of the new method’s theory

Epilogue | 161

and turn the mundane and painstaking research process into a joyful and passionate scientific research journey. New progress has been made in recent years with the guidance of applications of new ideas that have been the source of the Ramsey number brochures. It has shown that young scientists have become the major force in Ramsey theory research, and they continue to introduce new and more excellent results. Haipeng Luo Nanning March 2017

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Index A almost regular 21 arrow 3 B Banach–Mazur distance 154 bi-color diagonal classical Ramsey number 17 bipartite Ramsey number 108 bridge 73 C chromatic number 2 clique number 2 combinatorial line 123 complement 1 composition 2 connected Ramsey number 73 connectivity 2 cyclic graph 3 cyclic Schur number 126 D density 123 directed Folkman numbers 103 directed Ramsey number 103 disjoint Golomb ruler 131 E edge Folkman number 77 Erdős–Hajnal conjecture 95 Erdős–Hajnal number 95 F Folkman Games 138 G generalized Riemann hypothesis 26 Golomb rectangle 134 Golomb Ruler 131 Green–Tao theorem 112 H Hadwiger–Nelson problem 157 Hales–Jewett number 123 Hales–Jewett theorem 123 Hamiltonian-connected 51 https://doi.org/10.1515/9783110576702-020

hypergraph Folkman number 88 hypergraph Ramsey numbers 104 I independence number 2 isomorphic 2 isomorphism 2 K k-universal 99 L least quadratic non-residue 26 lexicographic product of graphs 2 local Folkman numbers 144 local hypergraph Ramsey number 106 local k-coloring 141 local Ramsey numbers 141 local van der Waerden numbers 144 Lovász–Ramsey number 95 M multigraph Ramsey numbers 147 P Paley graph 25 partition regular 124 prime counting function 37 Prime number theorem 37 R Rado number 124 Ramsey graph 3 Ramsey graph game 137 Ramsey multiplicity 107 Ramsey number 3 Ramsey theorem 7 S Schur number 125 set-coloring Folkman numbers 149 set-coloring van der Waerden numbers 115 Sidon number 131 Sidon set 131 Sidon–Ramsey number 132 size Ramsey numbers 106 Szemerédi’s theorem 111

178 | Index

T tight cycle 105 totally connected 73 Turán graph 2 Turán number 2 Turán’s theorem 2

V van der Waerden game 139 vertex Folkman game 138 vertex Folkman number 77 W weak Schur number 125