Structure And Randomness In Computability And Set Theory 2020034089, 2020034090, 9789813228221, 9789813228238, 9789813228245

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Structure and Randomness in Computability and Set Theory

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b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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Structure and Randomness in Computability and Set Theory Edited by

Douglas Cenzer

University of Florida, USA

Christopher Porter Drake University, USA

Jindrich Zapletal

University of Florida, USA

World Scientific NEW JERSEY

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Library of Congress Cataloging-in-Publication Data Names: Cenzer, Douglas, editor. | Porter, Christopher (Christopher P.), editor. | Zapletal, Jindrich, editor. Title: Structure and randomness in computability and set theory / edited by Douglas Cenzer, University of Florida, Christopher Porter, Drake University, Jindrich Zapletal, University of Florida. Description: New Jersey : World Scientific, [2021] | Includes bibliographical references and index. Identifiers: LCCN 2020034089 (print) | LCCN 2020034090 (ebook) | ISBN 9789813228221 (hardcover) | ISBN 9789813228238 (ebook for institutions) | ISBN 9789813228245 (ebook for individuals) Subjects: LCSH: Set theory. | Computable functions. Classification: LCC QA248 .S892 2021 (print) | LCC QA248 (ebook) | DDC 511.3/22--dc23 LC record available at https://lccn.loc.gov/2020034089 LC ebook record available at https://lccn.loc.gov/2020034090 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/10661#t=suppl Desk Editors: Kwong Lai Fun/Vishnu Mohan Typeset by Stallion Press Email: [email protected] Printed in Singapore

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Preface

The goal of this collection of review chapters is to provide an in-depth overview of work in three areas that have emerged on the frontier of research in set theory and computability in recent years: (1) infinitary combinatorics and ultrafilters; (2) algorithmic randomness and algorithmic information; and (3) computable structure theory. The unifying themes of these areas are an emphasis on structure, randomness, and the interplay between them. Taking the above-listed three research areas in reverse order, these themes appear as follows: First, we have an emphasis on the use of the tools of computability theory to analyze the complexity of various mathematical structures in computable structure theory. Second, we can study the extent to which effective methods allow us to classify objects as random or nonrandom, a central theme in algorithmic randomness, or, in the case of algorithmic information theory, measuring the amount of algorithmic information in an object, placing it on a scale that at the lower levels corresponds to more structure, while at the higher levels corresponds to more randomness. Third, in infinitary combinatorics, one can study the extent to which order emerges from disorder, particularly in the case of Ramsey theory for infinite sets, v

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thereby yielding an interesting balance between structure and a certain kind of randomness. We now layout the specifics of the book. Part I “Infinitary Combinatorics and Ultrafilters” deals with a remarkably persistent theme in set theory. Nonprincipal ultrafilters on natural numbers are well-known to be difficult to analyze and treat in detail. At the same time, they are exceptionally useful in many directions, as one can see from the proof of van der Waerden theorem via the compactification of the semigroup of natural numbers or (more recently) Malliaris’s and Shelah’s proof of p = t. Over the years, set theorists isolated certain critical properties of ultrafilters which completely determine their combinatorial features. As the oldest example of such a critical property, consider selective ultrafilters: the ultrafilters U which, for every coloring c : [ω]2 → 2, contain a set homogeneous for such a coloring. It turns out that it is impossible to find further distinctions between selective ultrafilters using formulas from a certain broad syntactically identified class. In other words, if we know that a certain ultrafilter is selective, we probably know most of its other combinatorial properties as well. One way to precisely formulate this intuition is found in a result of Todorcevic: granted sufficiently large cardinals, every selective ultrafilter U is generic over the canonical model L(R) (the smallest model of ZF set theory containing all reals and all ordinals) for the partially ordered set P of all infinite subsets of ω ordered by inclusion. Thus, the study of combinatorial properties of U expressible in the generic extension L(R)[U] reduces to the study of the partial ordering P and is completely independent of U. The model L(R)[U] attracted plenty of attention over the last 30 years from authors such as Todorcevic, Di Prisco, Dobrinen, Paul Larson and Zapletal; it is one of the canonical and best understood objects in transfinite set theory. Given the success story of Ramsey ultrafilters, one can ask whether it is possible to find other properties playing similar

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critical role. The answer is affirmative and the list of such critical properties is continually growing. Most of the examples found so far can be restated in terms of partition calculus. For each such critical property φ, there is a definition of a partial ordering Pφ such that every ultrafilter satisfying φ is in fact generic over the model L(R) for the poset Pφ . This converts the study of an ultrafilter with the property φ to the study of the ordering Pφ . The comparison of the models L(R)[U] for ultrafilters U of various critical types then slowly dissects the unwieldy set of all ultrafilters into smaller, manageable units. This open-ended, bold program has seen continual progress over the years, and it is intimately connected with the tools of abstract partition calculus. Part I consists of two chapters. Chapter 1 “Topological Ramsey Spaces Dense in Forcings”, by Natasha Dobrinen, is an extensive survey of this area. Dobrinen frames this traditional field of inquiry using the theory of topological Ramsey spaces of Stevo Todorcevic, which support infinite-dimensional Ramsey theory similarly to the Ellentuck space. Each topological Ramsey space is endowed with a partial ordering which can be modified to a σ-closed “almost reduction” relation analogously to the partial ordering of “mod finite” on [ω]ω . Such forcings add new ultrafilters satisfying weak partition relations and have complete combinatorics. In cases where a forcing turned out to be equivalent to a topological Ramsey space, the strong Ramseytheoretic techniques have aided in a fine-tuned analysis of the Rudin–Keisler and Tukey structures associated with the forced ultrafilter and in discovering new ultrafilters with complete combinatorics. This original perspective allows her to organize the wealth of existing research in a particularly incisive way and to prove a number of new results as well. Dobrinen’s exposition provides a long dictionary of critical combinatorial properties of ultrafilters and pointers for extending this dictionary further. Readers interested in using techniques using topological Ramsey spaces to study ultrafilters with various partition relations should find Dobrinen’s survey to be instructive.

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Chapter 2 “Infinitary Partition Properties of Sums of Selective Ultrafilters”, by Andreas Blass, is more specific and deals with a particular pair of critical combinatorial ultrafilter properties. It concerns two kinds of ultrafilters on ω 2 , the first kind given by the sums of nonisomorphic selective ultrafilters that are indexed by another selective ultrafilter, and the second kind given by ultrafilters that are generic with respect to the forcing the conditions of which are subsets of ω 2 that have an infinite intersection with {n} × ω for infinitely many n ∈ ω. Although these two kinds of ultrafilters share a number of properties, such as being Q-points but not P-points and not being at the top of the Tukey ordering, they also differ in several respects, as only ultrafilters of the first kind are basically generated while only ultrafilters of the second kind are weak P-points. Blass first shows that the infinitary partition property of ultrafilters of the first kind is of the same strength with what has been previously shown about the infinite partition property of ultrafilters of the second kind. This, in turn, leads to Blass’s second main result, obtained via an application of complete combinatorics, that the two kinds of ultrafilters are the same when viewed in different models of set theory. Lastly, Blass uses this result to account for how both the similarities and differences between the two kinds of ultrafilter arise. Part II “Algorithmic Randomness and Information” concerns two different research strands, namely, the study of algorithmically random sequences under Turing reductions, and the study of effective notions of Hausdorff dimension defined in terms of Kolmogorov complexity, the central concept in algorithmic information theory. One of the primary aims in the study of algorithmic randomness is to study various definitions of algorithmically random sequences and the properties of such sequences. The most wellstudied definition of algorithmic randomness is Martin-L¨of randomness. Martin-L¨of’s original idea behind his definition was to formalize the notion of an effective statistical test, given

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by a sequence of effectively generated open sets the measures of which are effectively converging to zero. A sequence that is not in the intersection of any such test is a Martin-L¨of random sequence. Alternative definitions of algorithmic randomness can be obtained, for instance, by modifying the underlying notion of an effective statistical test, although Martin-L¨of has proven to be, in certain respects, more well-suited to being studied from a computability-theoretic point of view compared to alternative definitions of randomness. One respect in which Martin-L¨of randomness is amenable to study using tools from computability theory, namely the behavior of random sequences under Turing reductions, is the subject of the first randomness-theoretic chapter in this book, “Limits of the Kuˇcera–G´acs Coding Method”, by George Barmpalias and Andrew Lewis-Pye. This chapter discusses an improvement of methods used independently by Kuˇcera and G´acs to prove what is now considered to be a classical result in the field concerning the computational power of Martin-L¨of random sequences. In 1985, Kuˇcera proved that for every sequence A ∈ 2ω , there is some Martin-L¨of random sequence B such that A ≤T B. Kuˇcera’s proof involved a method of coding according to which one encodes an arbitrary sequence A ∈ 2ω by a member of a fixed Π01 class of positive Lebesgue measure. If we use this method of coding on a Π01 class consisting of Martin-L¨of random sequences (such as the complement of some level of the universal MartinL¨of test), we obtain the desired reduction to a random sequence. Independently, in 1986 G´acs proved the same result using an alternative method of coding, which allowed him to provide a more fine-grained analysis of the reduction in question, which he shown can be given by a wtt-functional. That is, G´acs proved that there is a computable function f : ω → ω such that every sequence A is computable from a Martin-L¨of random sequence B with the function f bounding the use of this reduction. In fact, G´acs proved that we can take this function to be given by √ f (n) = n + n log(n).

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In their chapter, Barmpalias and Lewis-Pye first carefully layout a modular argument for the theorem that encompasses the coding methods employed by both Kuˇcera and G´acs. Next, they identify a key limitation of this approach which involves the redundancy of the reduction, i.e., which, for each n, is the number of additional input bits beyond n that are needed to yield n output bits (for instance, the proof√due to G´acs establishes the result with redundancy g(n) = n log(n)). The authors then provide an alternative coding method, which allows them to prove that every sequence can be computed from a MartinL¨of random sequence with an optimal logarithmic redundancy, thereby significantly improving the original Kuˇcera–G´acs result. One of the most useful tools in the study of algorithmic randomness, namely Kolmogorov complexity, can be used to measure the amount of algorithmic information in a given sequence (for this reason, this area of research is sometimes referred to as algorithmic information theory). Informally, the Kolmogorov complexity of a finite string is the number of bits required of a universal machine to output that string; in the case that the Kolmogorov complexity of a string exceeds the length of a string, such a string is called incompressible (it is standard to consider incompressibility up to a fixed additive constant, but we will suppress that detail here). One can then consider the Kolmogorov complexity of the initial segments of an infinite sequence. It was recognized fairly early in the development of the theory of algorithmic randomness that Martin-L¨of random sequences are precisely those sequences with sufficiently incompressible initial segments (at least with respect to certain modifications of the originally formulated notion of Kolmogorov complexity, such as prefix-free complexity and monotone complexity). It was later discovered that studying the initial segment complexity of initial segments of nonrandom sequences can still be quite fruitful. This leads to the following chapter, “Information vs. Dimension: An Algorithmic Perspective”, by Jan Reimann, which is a self-contained survey that covers the work on the

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interactions between algorithmic information theory and various notions of fractal dimension dating back over 30 years. Informally, the fractal dimension of a set, for instance a subset of R2 , measures the complexity of the set in a way that corresponds to the extent that it fills spaces. Moreover, fractal dimension is not integer-valued, as it can provide intermediate values between the standard spatial dimensions. Beginning in the 1980s, results due to Ryabko and Staiger indicated a strong connection between the classical notion of the Hausdorff dimension of a subset of a sequential space and the Kolmogorov complexity of initial segments of its members. Lutz later extended this work by defining effective notions of Hausdorff dimension and packing dimension that are defined for individual binary sequences, a development that marked a departure from the classical versions of these notions, which always assign dimension zero to an individual object. These definitions of effective dimension proved to be robust: Lutz’s definitions of these two notions of dimension were initially given in terms of certain betting strategies known as gales, but it was later shown by various authors that these two notions can be given formulated in terms of Kolmogorov complexity: the effective Hausdorff dimension of an infinite sequence X is the limit infimum of the ratio K(X  n)/n (where K(X  n) is the prefixfree Kolmogorov complexity of the first n bits of the sequence X), while the effective packing dimension of X is the limit supremum of this same ratio. These developments, as well as more recent developments, are covered in Reimann’s survey. After reviewing the basics of classical information theory, Kolmogorov complexity, and Hausdorff dimension, Reimann details various aspects of effective Hausdorff dimension, including its formulation in terms of effective null sets and its relationship to Kolmogorov complexity. He then highlights the role effective Hausdorff plays in the effective version of the Shannon–McMillan–Breiman theorem from classical information theory, a result that underwrites the informal

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identity “entropy = complexity = dimension”. Finally, Reimann discusses some of his own results on multifractal measures that draw upon effective Hausdorff dimension. Part III, “Computable Structure Theory”, contains three chapters. There are some unifying themes. One is definability in the arithmetic and hyperarithmetic hierarchy, that is, the effective Borel hierarchy. A second theme is computable reducibility. This leads to notions of completeness of sets and relations at various levels of the hierarchy. Index sets are used to characterize properties of structures frequently by showing that the index sets are complete at a certain level. Chapter 5 by Russell Miller “Computable Reducibility for Cantor Space” is a contribution to the quickly growing field of computable descriptive set theory, evaluating the computational content of various objects and relations appearing in classical descriptive set theory. The field identifies distinctions which are too fine for descriptive set theory to detect, and opens a new perspective in each major theme of descriptive set theory. Descriptive set theorists often build hierarchies of objects found in mathematical analysis. There is the Wadge hierarchy of definable subsets of the Baire space, which is in a certain direction the ultimate complexity measurement. However, one can find other similar tools with different aims. Large parts of several decades of research in the theory of cardinal invariants of the continuum can be viewed as the study of σ-ideals on Polish spaces ordered by Borel–Katˇetov reductions. A somewhat younger research direction in similar vein is the comparison of Borel equivalence relations via Borel reducibility. In this case, a detailed map of Borel equivalence relations has been drawn, encompassing and rating many known equivalence problems. This has been a very successful enterprise; several long-standing classification research programs in mathematical analysis went down in flames after it was shown that they can never capture the inherent complexity of the underlying objects.

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As soon as workers in computability found a satisfactory notion of a computable function between Polish spaces, they started wondering how complex the reduction functions in these descriptive set theoretic hierarchies must be from computability standpoint. A parallel to the theory of cardinal invariants was discovered in the study of Turing degrees, and interesting finesses appeared in the computational complexity of Katˇetov reductions between various σ-ideals. The theory of computable reductions between Borel equivalence relations has not received as much attention, and Miller’s paper aims to rectify this situation. Miller surveys a number of possible ways of restricting the comparison to computable reducibility of equivalence relations on the Cantor space. Different versions are obtained by modifying some aspect of the reduction. For instance, one can vary the number of jumps of the input sequence that the functional is allowed to use, one need not require that the reduction succeed for all members of Cantor space but only for arbitrary finite or countable subsets of Cantor space, or one can add an additional oracle to be used in the reduction. As Miller shows, these refinements of Borel reducibility allow one to classify Borel reductions in terms of a level of difficulty or to account for why a Borel reduction fails to exist. Chapter 6 “Logic Programming and Effectively Closed Sets”, by Douglas Cenzer, Victor W. Marek and Jeffrey B. Remmel, surveys results on index problems for effectively closed sets and their applications to models of logic programs. Effectively closed sets are the foundational level of the effective Borel hierarchy. They play a key role in the area of algorithmic randomness. Effectively closed sets, or Π01 classes, arise naturally in the study of computable structures. For many mathematical problems, such as finding a prime ideal of a Boolean algebra or finding a zero of a continuous function, the set of solutions to a given problem may be viewed as a closed set under some natural topology. For a computable problem, the set of solutions may be

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viewed as a Π01 class. Thus, completeness results for properties of Π01 classes may be used to obtain completeness results for many types of computable problems. For example, the set of complete consistent extensions of an axiomatizable theory may be represented as a Π01 class, and the property of having a computable complete consistent extension can be shown to be Σ03 complete. The authors introduce a new notion of boundedness for trees and examine the complexity of index sets for the corresponding closed sets. This classification is in turn used to study the so-called recognition problem in the metaprogramming of finite normal predicate logic programs. In particular, for any property P of finite normal predicate logic programs over a fixed computable first-order language, the associated index set IP is classified in the arithmetical hierarchy. Here, the authors’ classification of the index sets of closed sets associated with the abovedescribed bounded trees serves as the primary tool. For example, the authors determine the complexity of the index sets relative to all finite predicate logic programs and relative to certain special classes of finite predicate logic programs of properties such as (i) having no stable models, (ii) having no recursive stable models, (iii) having exactly c stable models for any given positive integer c, (iv) having exactly c recursive stable models for any given positive integer c, (v) having only finitely many stable models, and (vii) having only finitely many recursive stable models. Chapter 7 “Computability and Definability”, by Trang Ha, Valentina Harizanov, Leah Marshall and Hakim Walker provides a comprehensive survey of recent results in computable structure theory, including much recent work that has not been covered in previous surveys on the field. One important theme is degree spectrum of a structure and of a theory. Another key topic is the notion of a Scott sentence of a structure, in particular for finitely generated structures. Other topics covered include

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effective aspects of theories, diagrams, and models, computable infinitary formulas and Scott rank, the index sets of structures and classes of structures, strongly minimal structures, relatively Δ0α -categorical structures, and aspects of definability and complexity of relations on structures. Particular structures studied include graphs, trees, linear orders, groups and fields. Recent results of Knight, Harrison-Trainor and others characterize those structures which have a d-Σ02 Scott sentence and establish that any finitely generated field has such a sentence. Computable structures of Scott rank ω1CK lead up to the solution by Harrison-Trainor, Igusa and Knight, of a long-standing question of Sacks by showing that there such a computable structure such that the computable infinitary theory is not ℵ0 -categorical extensive treatment of computable and Δ0α categoricity includes characterization of such categoricity in terms of Scott families; the distinction between Δ0α categoricity and relative categoricity. Results of Downey, Lempp, Montalban, Turetsky and others show that the index set for computably categorical structures is shown to be Π11 complete whereas the index set for relatively computable structures is only Σ03 complete. There are new results of Adams and Cenzer on weakly ultrahomogeneous structures. There is a detailed presentation of the work of Hariizanov, Knight and others on intrinsically Σ0α relations. This includes very interesting new results about the degree spectrum of a relation. Index sets for classes of structures are examined in detail. For example, the index set of computable prime models is Π0ω+2 -complete, the index set of computable structures with noncomputable Scott rank is Σ11 -complete, and the set of finitely generated free groups is Σ03 complete within the class of all free groups. The topics covered in this book were inspired by a sequence of meetings in Gainesville, FL, known as the Southeastern Logic Symposium (SEALS). Experts from computability and set theory have been meeting since 1985 at a number of institutions

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in the Southeast US, with support from the National Science Foundation. The editors would like to thank the authors and the referees for their hard work and patience. Douglas Cenzer, Chris Porter and Jindra Zapletal Gainesville, FL

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About the Editors

Douglas Cenzer is Professor of Mathematics at the University of Florida, where he was Department Chair from 2013 to 2018. He has more than 100 research publications, specializing in computability, complexity, and randomness. He joined the University of Florida in 1972 after receiving his Ph.D. in mathematics from the University of Michigan. Christopher Porter is Assistant Professor of Mathematics at Drake University, specializing in computability theory, algorithmic randomness, and the philosophy of mathematics. He received his Ph.D. in mathematics and philosophy from the University of Notre Dame in 2012, was an NSF international postdoctoral fellow at Universit´e Paris 7 from 2012 to 2014, and a postdoctoral associate at the University of Florida from 2014 to 2016, before joining Drake University in 2016. xvii

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Jindrich Zapletal is Professor of Mathematics at University of Florida, specializing in mathematical logic and set theory. He received his Ph.D. in 1995 from the Pennsylvania State University, and held postdoctoral positions at MSRI Berkeley, Cal Tech and Dartmouth College, before joining the University of Florida in 2000.

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Contents

Preface

v

About the Editors

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Part I: Infinitary Combinatorics and Ultrafilters

1

1. Topological Ramsey Spaces Dense in Forcings

3

Natasha Dobrinen 2. Infinitary Partition Properties of Sums of Selective Ultrafilters

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Andreas Blass

Part II: Algorithmic Randomness and Information

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3. Limits of the Kuˇ cera–G´ acs Coding Method

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George Barmpalias and Andrew Lewis-Pye xix

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4. Information vs. Dimension: An Algorithmic Perspective

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Jan Reimann

Part III: Computable Structure Theory 5. Computable Reducibility for Cantor Space

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Russell Miller 6. Logic Programming and Effectively Closed Sets

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Douglas Cenzer, Victor W. Marek and Jeffrey B. Remmel 7. Computability and Definability

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Trang Ha, Valentina Harizanov, Leah Marshall and Hakim Walker Author Index

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Subject Index

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Infinitary Combinatorics and Ultrafilters

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Chapter 1

Topological Ramsey Spaces Dense in Forcings Natasha Dobrinen Department of Mathematics, University of Denver, C.M. Knudson Hall 302, 2290 S. York St., Denver, CO 80208, USA [email protected] http://cs.du.edu/∼ndobrine Topological Ramsey spaces are spaces which support infinite dimensional Ramsey theory similarly to the Ellentuck space. Each topological Ramsey space is endowed with a partial ordering which can be modified to a σ-closed “almost reduction” relation analogously to the partial ordering of “mod finite” on [ω]ω . Such forcings add new ultrafilters satisfying weak partition relations and have complete combinatorics. In cases where a forcing turned out to be equivalent to a topological Ramsey space, the strong Ramsey– theoretic techniques have aided in a fine-tuned analysis of the Rudin–Keisler and Tukey structures associated with the forced ultrafilter and in discovering new ultrafilters with complete combinatorics. This expository paper provides an overview of this collection of results and an entry point for those interested in using topological Ramsey space techniques to gain finer insight into ultrafilters satisfying weak partition relations.

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Overview

Topological Ramsey spaces are essentially topological spaces which support infinite-dimensional Ramsey theory. The prototype of all topological Ramsey spaces is the Ellentuck space. This is the space of all infinite subsets of the natural numbers equipped with the Ellentuck topology, a refinement of the usual metric, or equivalently, product topology. In this refined topology, every subset of the Ellentuck space which has the property of Baire is Ramsey. This extends the usual Ramsey Theorem for pairs or triples, etc., of natural numbers to infinite dimensions, meaning sets of infinite subsets of the natural numbers, with the additional requirement that the sets be definable in some sense. Partially ordering the members of the Ellentuck space by almost inclusion yields a forcing which is equivalent to forcing with the Boolean algebra P(ω)/fin. This forcing adds a Ramsey ultrafilter. Ramsey ultrafilters have strong properties: They are Rudin–Keisler minimal, Tukey minimal, and have complete combinatorics over L(R), in the presence of large cardinals. These important features are not unique to the Ellentuck space. Rather, the same or analogous properties hold for a general class of spaces called topological Ramsey spaces. The class of such spaces were defined by abstracting the key properties from seminal spaces of Ellentuck, Carlson–Simpson, and Milliken’s space of block sequences, and others. Building on the work of Carlson and Simpson [8], the first to form an abstract approach to such spaces, Todorcevic presented a more streamlined set of axioms guaranteeing a space is a topological Ramsey space in [35]. This is the setting that we work in. Topological Ramsey spaces come equipped with a partial ordering. This partial ordering can be modified to a naturally defined σ-closed partial ordering of almost reduction, similarly to how the partial ordering of inclusion modulo finite is defined from the partial ordering of inclusion. This almost reduction

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ordering was defined for abstract topological Ramsey spaces by Mijares in [27]. He showed that forcing with a topological Ramsey space partially ordered by almost reduction adds a new ultrafilter on the countable base set of first approximations. Such ultrafilters inherit some weak partition relations from the fact that they were forced by a topological Ramsey space; they behave like weak versions of a Ramsey ultrafilter. When an ultrafilter is forced by a topological Ramsey space, one immediately has strong techniques at one’s disposal. The Abstract Ellentuck Theorem serves to both streamline proofs and helps to clarify what exactly is causing the particular properties of the forced ultrafilter. The structure of the topological Ramsey space aids in several factors of the analysis of the behavior of the ultrafilter. The following are made possible by knowing that a given forcing is equivalent to forcing with some topological Ramsey space. (1) A simpler reading of the Ramsey degrees of the forced ultrafilter. (2) Complete Combinatorics. (3) Exact Tukey and Rudin–Keisler structures, as well as the structure of the Rudin–Keisler classes inside the Tukey classes. (4) New canonical equivalence relations on fronts — extensions of the Erd˝os–Rado and Pudl´ak–R¨odl Theorems. (5) Streamlines and simplifies proofs, and reveals the underlying structure responsible for the properties of the ultrafilters. This chapter focuses on studies of ultrafilters satisfying weak partition relations and which can be forced by some σ-closed partial orderings in [10, 11, 13, 15, 16], and other work. These works concentrate on weakly Ramsey ultrafilters and a family of ultrafilters with increasingly weak partition properties due to Laflamme in [23]; P-points forced n-square forcing by Blass in [3] which have Rudin–Keisler structure below them a diamond shape; the k-arrow, not k + 1-arrow ultrafilters of Baumgartner

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and Taylor in [1], as well as the arrow ultrafilters; new classes of P-points with weak partition relations; non-P-points forced by P(ω × ω)/(Fin ⊗ Fin) and the natural hierarchy of forcings of increasing complexity, P(ω α )/Fin⊗α . It turned out that the original forcings adding these ultrafilters actually contain dense subsets which form topological Ramsey spaces. The Ramsey structure of these spaces aided greatly in the analysis of the properties of the forced ultrafilters. In the process some new classes of ultrafilters with weak partition properties were also produced. Though there are many other classes of ultrafilters not yet studied in this context, the fact that in all these cases dense subsets of the forcings forming topological Ramsey spaces were found signifies a strong connection between ultrafilters satisfying some partition relations and topological Ramsey spaces. Thus, we make the following conjecture. Conjecture 1. Every ultrafilter which satisfies some partition relation and is forced by some σ-closed forcing is actually forced by some topological Ramsey space. While this is a strong conjecture, so far there is no evidence to the contrary, and it is a motivating thesis for using topological Ramsey spaces to find a unifying framework for ultrafilters satisfying some weak partition properties. Finally, a note about attributions: We attribute work as stated in the papers quoted. 2.

A Few Basic Definitions

Most definitions used will appear as needed throughout this chapter. In this section, we define a few notions needed throughout. Definition 2. A filter F on a countable base set B is a collection of subsets of B which is closed under finite intersection and closed under superset. An ultrafilter U on a countable base set

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B is a filter such that each subset of B or its complement is in U. We hold to the convention that all ultrafilters are proper ultrafilters; thus ∅ is not a member of any ultrafilter. Definition 3. An ultrafilter U on a countable base set B is (1) Ramsey if for each k, l ≥ 1 and each coloring c : [B]k → l, there is a member U ∈ U such that c  [U]k is constant. (2) selective if for each function f : ω → ω, there is a member X ∈ U such that f is either constant or one-to-one on U. (3) Mathias-selective if for each collection {Us : s ∈ [ω]