Surveys in Combinatorics 1987: Invited Papers for the Eleventh British Combinatorial Conference 0521348056, 9780521348058

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Londen: Meihematical Society Lecture Note ReneS nO

CAMBRIDGE

UNIVERSITY

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LONDON MATHEMATICAL SOCIETY LECTURE NOTE'SERIES Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England The books in the series listed below are available from booksellers, or, in case of difficulty,

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London Mathematical Society Lecture Note Series. 123

Surveys in Combinatorics 1987 Invited papers for the Eleventh British Combinatorial Conference

Edited by C. WHITEHEAD Department of Mathematical Sciences, Goldsmiths' College, London

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© Cambridge University Press 1987 First published 1987 Printed in Great Britain at the University Press, Cambridge Library of Congress cataloging in publication data available British Library cataloguing in publication data British Combinatorial Conference (11th: 1987: Goldsmiths’ College) Surveys in combinatorics, 1987: Invited papers for the Eleventh British Combinatorial Conference. —- (London Mathematical Society lecture note series, ISSN 0076-0552; 123).

1. Combinatorial analysis I. Title 511'6

ISBN

I. Whitehead, C. QA164

0521 34805 6

III. Series

CONTENTS Preface Barlotti, A. Finite geometries and designs

Cameron, P.J. Portrait of a typical sum-free set

13

Chvatal, V. Perfect graphs

43

Erdés, P. My joint work with Richard Rado

53

Frankl, P. The shifting technique in extremal set theory

81

Graham, R.L. & Rédl, V. Numbers in Ramsey theory

111

Milner, E.C. & Prikry, K. Almost disjoint sets

155

graphs Thomason, A. Random graphs, strongly regular and pseudo-random graphs

173

y Winkler, P. The metric structure of graphs: theor and applications

197

Index of names

223

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an

see

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and

new

passes

a ana

exactly

following

incidence

is

longer

(when

are

by simply

a

plane

To

Finite

modification

the

lh

The

the

The

affine

S-spaces.

points

through

satisfied.

through in

is

line

an

of dimension

in

and

satisfied.

space

lie

of

points,

is

I'

way:

of

suitable

incidence

of

is not

L,

which

through

are

a

and

notion

of

example

S(P,

the

one

parallelism.

two

a line

to

from

following

AB),

with

Z

lines

pass

S-spaces

but

the

number

space

in

the

class

affine

S-space

the

or

easy

special

simple

ordinary

exactly

a.

arise

@

exists

generalizes a

very

a plane

A

Veblen

a through

A

in

change,

in

may

relation

for

and

of

axioms:

number

there

designs

an

Similarly

It postulates

be

to

form

S-spaces

2 except

B.

exactly

consists

elements

PXxL

with

2-(v,k,1)

an

through

in

of

following

a finite

obtain

a

L

the

with

spaces.

through

set

cardinal

a)

S-space

S-spaces

called

not

A with

(A,

spaces

can

in

pair

affine is

incidence

a

S,

relation;

an

Consider

2%, we

S-space,

defined

same

of

affine

s>3.

the

notion

Ete ae, Bh)

From

with

equivalence

A

Proper

order

I

incident

parallel

structures

Let

are

and

that

ordinary

and

satisfying

points

an

with

The space,

L,

point-line

incident

points

an

5S;

3) Parallelism

4) For

shortly

relation

L*

incident

of

or

called

in

distinct

order

space

incidence

defined

1) Any

the

affine

be

two

the

desarguesian

affine r

line

of

2),

Then

lines

space

bea

a line A.

four

rand

of

8 not

Barlotti: Finite geometries and designs

through In

A

the

and

which

S-space,

u.

the

Thus

satisfied.

the

Barlotti

& Nicoletti

may

1968)

spread

be

very

and

of

construction

of

planes

in

use

a later

Z and

Let

q.

2 be

two

of

= .

plane

am

intersection

A-points

them.

represented

than

by the

and

s

represented

Q , and

A

avoid

to

A-lines

by

the

in

useful

high

the

to

also

may

products

Similar

other

of

types

several

of

the

as a natural

arises

and

1979).

geometry

(see,

construction

of

structures.

given

in Bose &Barlotti

Dembowski

e.g.

non-desarguesian

the

here

recall

We

incidence

are

and

A-points

points them

confusing

three

of

are

of

planes

planes

and

£

¥!

contained passing

of type

Q

and

.

The

lines

with

shall

1971,

which

we

4-dimensional

space

of order

which

in

to be

plane

will

5

of

line

constructed

A-points

type

A-points

of

pass

through

a line

Eu

A—points:jof

s

and

represented

not

lying

by the

of

“type

(1) S

(2)

in either

points

will

and

represent

which

2!

a

in the

contained

be called of

by S

denote

and

%'

of

elements

types.

through

(3) are

the

s

by

Denote

its

not

3-spaces

distinct

of

finite

a

be

PG(4,q)

=

lines

by

e.g.

(see

section.

of

denoted

S-spaces

orthographic

the

to

1977).

Halder

1978,

projective

a

of

A-planes

spread

be

in

non—desarguesian

extended

be

construction

Glynn

other

"

the

choosing

in

a

classes

has

designs

of

order

a and

the

allows

semitranslation

of

the

over

their

spread

constructed

other

spreads

in if

a

line.

contained

ring

only

in AM

of

(1)

and

“A—point.”

ternary

or

1s

this

a

plane

& of

in

PG(2n,p)

from

(Krier

has

“to be

been

way of

is

dimensional

A

given

in

of

linear

Heft

of

the

27.

the

design. of

1971

and

in)

classes spaces

obtain

different

X'

flats

By

planes

1974).

a divisible

various

we

shear

hyperplanes

(n-1)-dimensional

in this

high

Q

generalizations.

derived

A

constructions in

is

various

kernel

of

2 and

Galois

of type

A-point

“the®

of

planes

construction

PG(2n,q), a

a

as

1971.

way

2

defined

intersect

A-line

representing

dimension

be

using

12!

with

is

A-point

them

the

the

than

given

of

& Smith

structure

also

incident

suitable

i=

incidence

is

A

of

to

representing

of

generalization

so

planes

construction

greater

taking

is incident

representing

plane

in Bose

(2)

A-line

element

The

spread

the a

is obtained

by

if

case

semi-translation

degree

type

other

contains

other

of

between

Heft

partial

ani on

certain

quadrics.

4.

a

variety

Well

known

Geometrical

in

a

1981,

briefly

unital

the

arc)

have

examples

Ray-Chaudhuri Holz

space

1962,

properties

are

in

1965.

Pellegrino

the

plane

used

Bose

Other

1973,

construction

in

been

of

A

a

curve

for

the

1958,1961,

examples

1974, a

of

(Barlotti

of

Bose

&

given

1978.

lines

finite

construction

are

1977,

unital

ina

We

plane

of

or

of

designs.

Chakravarti

1966,

in

1968,

Barlotti

shall

recall

here

U*

(i.e.

the

dual

& Lunardon

1979).

The

existence

of

a

Barlotti: Finite geometries and designs

of,

1%

implies

consists

of

of

U*.

the

The

s

in

§ 3.

2

in

two

A

there

points

of

A

which

A

Let

Q

of

is

_ Barlotti

& Lunardon

In Biscarini

designs.

is

other

Among be and

is

for

used vifs.

also

For

proved

6.

the

is

E'

in

the

plane

U*

of

P and assume

with

two

of

U*.

Let

in P.

from

by

the

set

(1)

type

possible

all

and

Consider

A

is a unital

A-lines

consider

to

2%

obtained

passing

incident

and

and

of

points

&

that

1

q+

only

of

of

&

different

cone

the

line

1976.

points

V

lines

A-point

every

exactly

a Laguerre

design

with

the

construction of

how

2-designs

Another

but not

structures

incidence

designs

of

geometric

we

be

associated

way

which

plane

do

C which

the

through

q+1

and

lines

it

A-lines

of

of

not

is

This

(cf.

cases

in the

point

out

see

Glynn

to

of

to

of

q

order

PG(3,q)

PG(3,q).

finite the

designs.

interesting

of odd

parameters

isomorphic

which

structures

these

can

same

class

construct

to

from

and planes)

theory

used

1980

& Conti

the

be

may

known

a well

are

planes

planes

Benz

of points

(as a structure

prove

to

order

the

only

q

qe -

of

in

A

planes

the

by

or

w

intersects

by

either

%'

in

,

P

Consider

V.

2

with

(3-dimensional)

point

,

respectively.

(clearly

&

with

p

p

that

way

a

w

one

intersects

incident

symmetric

a

constructed

which

S

line

exactly

Z'

of

the

inversive

finite

of

types

2'

which

1979).

Finite

5.

using

of

arc

in Buekenhout

and

requires

and

difficult

Other

with

incident

is

U*

constructed

given

oO

the

that

prove

to

enough

in

(2),

type

of

lines

that

to

consists

U*

Since

VR.

line

to

with

is tangent

of

,

XYUQ

to

belong

not

incident

unital

and

represented

lines

of

a

oOo

such

the

from

also’

denoted

be

C

by

Q

projecting U*

p

in

are

a hyperplane

planes

Let

denote

and

P),

be

exists

analogous

plane

intersection

the

be

V

A

chosen

is

A

the

A _ which

respectively.

that

is

be

distinct

ovoid

R

in

construction

Let

an

that

case

may

half-planes

1985,

where

leads

to

it

them.

sometimes

the

Barlotti: Finite geometries and designs

construction

covering chosen A

or

in

and

of

B

families

of

a

a proper

the

designs

partition way.

of

We

respectively,

design,

of

incidence

fix

the

in

be

based

a geometric

may

as

may

two

set

of

the

on

structure

particular blocks

design

the

and

being

knowledge

with

set

given

a

substructures

families the

of

of

of

by

elements,

points

a

of

a

particular

relation. We

1984.

The

plane

in

a

present

here

author

first

Lemma:

Consider

tangent

to

PG(3,q) \\%=

one

proves

Q

of

the

in

nice

point

aset

examples

given

in

Vecchi

following.

PG(3,q)

ina

AG(3,q)

the

V.

D

(q

odd),

Then

it

of

a is

Re

wee

possible

Qs

to

anda

construct

q _ disjoined

fa cae

such

a

of

points

that

(i) gN\tvie bp (ii)

the

caps

of

D

form

partition

the

of

AG(3,q). From

PG(3,q) as

this

(q odd)

points

the

a block

are

meet

two

in

Cofman

Ebert

set

if

and

only

PG(3,q)

using

examples are

The

it

indefinite

of

When

Combinatorics

and

thinking

of

are

in

(q+1)-arcs

I since I

was at

another

a design

time

in

can

a,

a point

and

and

line

a family 428-429

pp.

planes

and

other

obtained

obtained

1986

designs

and

3-designs.

recently:

by

of

in

partitioning

are

constructed

odd).

this

busy free

for

with

1969

be

I was

possibility

to

ees

Bruck

present

remain

that

of

parameters:

of a line

constructions

to

lines

» A =a° (+1) (q-3)/4.

in PG(2,q)(q

to

with

the

belonging

corresponding

Kestenband

that

those

the

a covering

designs

invited

wanted

and

blocks

establishing

inversive

nice

as

and

obtain

on

how

and

if

V_

considered

two

shown

ovoids,

title

also

those

D,

Taking

r=q°(q°-1)/2

based

recall is

properties

7.

we

results

will

into

points,

» ke(q-1)/2-,

points

we

through

the

distinct

1985

those

of

1973.

Finally

Theorem:

incident

Other of

the

Sees

v=q ; b=q°(q+1) sets

follows

, except

lecture

with to

this

I

gave

problems

decide.

lecture

far

Then

in

an from

I

was

order

to

Barlotti: Finite geometries and designs

of

keep topic

class

of

know

everything

here

the

1982,

following

Piper

divisible

of

1985.

Schulz

cf.

designs

translation

to

classified

be

classification

a

For

principles.

different

wish

1983,

Kelly

good

I present

designs

may

designs

of

types

Other

1967.

Schulz

1983,

who

those

For

a,b,1985,

1984

Butler

information:

necessary

a particular

symmetric

of

classification

the

on

designs.

symmetric

, the

g.

e.

designs,

in

designs,

of

types

choose

to

have

we

classification

a nice

have

of

variety

large

the

to

due

Clearly, to

the

for

above.

presented

order

to

better

was

I decided

and

anniversary

50-th

the

for

celebration

the

it

that

thought

I

later

However

planes.

projective

a classification

of

birth

second

the

of

anniversary

30-th

the

celebrate

REFERENCES

Barlotti,

A.

Alcuni

(1965).

di

generalizzati

Barlotti,

pp.

A.

(1968).

and

planes

Periodico

Barlotti,

A.

(1985).

designs.

Barlotti,

A. from

&

geometric

affini

Univ.

Padova

and

Geometric

Ars

Combin.

J.

46 pp.

(4)

77

la

costruzione

(1975),

for procedure pp. 7-14.

from

pp. the

projective

of

28-38.

(1974). and

40,pp.

231-241.

Finite

affine

Sperner

spaces.

Abh.

spaces

constructed Sem.

Math.

nei

unitals A. & Lunardon, G. (1979). Una classe di 5. 781-78 Riv. Mat. Univ. Parma (4) 5, pp.

Barlotti,

G. (1976). On a geometric procedure A. & Nicoletti, r spaces. Abh. Math. Sem. Univ. Sperne of on constructi 251-255.

of

construction

19-A,

projective

pp.

spaces.

projective

Barlotti,

45,

di

59-68.

construction

structures

Math.-Verein.

Cofman,

Hamburg

di Mat.

Representation other

Deutsch.

Jber.

spazi

Mat.

per

geometrici

Procedimenti

(1975).

A.

Rendiconti

Sperner.

degli

Sem.

18-46.

35,

PBIB-—disegni.

Barlotti,

studio

nello

risultati

Univ.

A-piani.

for

the

Hamburg

Barlotti: Finite geometries and designs

Barnabei,

M.

&

Brini,

A.

Beth,

Th.,

(1980).

Jungnickel,

D.

&

A.

(1982).

Lenz,

A.

(1983).

P.

&

Bose,

R.

C.

(1947).

The

Bose,

R.

C.

(1958).

deriving

ts.

The

Soc. Bose,

R.

On

for

C.

die

in die

(1980).

Sem.

Fis.

design

of

the

Design

endliche

endliche

Piani

Univ.

di

Modena

Theory.

Geometrie

Geometrie

Laguerre 2opepp.)

experiments,

application

a certain

golden

pp.

R.

C.

B.I.

II.

B.I.

Proc.

e

disegni

LO1=107i

34th

Indian

of

finite

projective

geometry

series

of balanced Kirkman arrangemen— commemoration volume, Calcutta Math.

jubilee

Lecture

& Barlotti,

projective

Bose y Rn

Ccn

18

di

Notes.

A.

Dept.

(1971).

planes Mat.

Pura

Chakravarti,

finite

of

Statistics.

in

a

e Appl.

I.

projective

Linear four

Chapell

M.

representation

dimensional

88,

pp.

(1966).

space

PG(N,q

Rie

iC

ak

Smiths,

yKey

JieeeC

(1971).

representable

Convegno

Geometria

(1969).

planes,in pp.

Buekenhout,

F. planes pp.

Construction Univ.

(1976). of

189-194,

varieties

Canadian

Ternary

order

of

J.

in

Math.

q

problems

with

of

Carolina

unitals

a kernel

of

of

of

finite its

in order

class

In

Universita

and Press,

a

planes.

Perugia

Mathematics

North

Existence

rings

semi-translation

Combinatoria,

"Combinatorial

426-514.

of

space.

a 18,

;

linearly

R. H.

of a class

projective

Hermitian ).

69-101.

Bruck,

Hill,

9-32.

pp.1161-1182. Bosies

Sci.

341-354.

(1961).

Annali

B.I.

I.

NiciCes

Bose,

from

1-25.

pp.

Congress,

obtained

13-17.

pp.

Mannheim.

F.

Atti

29,

(1985).

in

Einfiihrung

Conti,

simmetrici.

PBIB-designs

Mannheim.

Wissenschaftsverlag,

Biscarini,

H.

Einfiihrung

Wissenschaftsverlag,

Beutelspacher,

of

Math.

Mannheim.

Wissenschaftsverlag,

Beutelspacher,

A class

Discrete

geometries.

projective

pp.

projective

applications",

Chapel

finite q,

of

atti

Hill,

N.C.

translation Geom.

Ded.

5;

Barlotti: Finite geometries and designs

Butler,

N.

T.

(1984a).

pp.

Butler,

N.T.

(1984b).

Giessen,

Butler,

N.T.

Clatworthy,

W.

(1954).

On

combinatorics Projective

Halder,

Hedayat,

5S.

Ph.D.

An

(1985).

309-342.

H.-R.

(1979).

A.

J.

M.

G.

type

V and

Math.

Sem.

VI.

J.

Statist.

configuration Am.

Math.

finite

Soc.

projective

pp.

59-70,

which

is

pp.

47-55.

5,

a

spaces.

Proc.

Washington

State

Heidel-

Berlin

caps.

into

geometries

projective

1163-1175.

of Adelaide.

Paralle-

mit

von Inzidenzrdiumen Produktbildung Angew. Math. 305, pp. 82-88.

19A,

Combin.

Ars

planes,

half

to

introduction

combinatorial

and related

planes

Univ.

Thesis,

S.

Spreads

Ph.D.

The

(1980).

(1971).

Inference

projective

in

Chapel

Thesis,

von

Konstruktion

projektiven

pp.

Hill,

t-designs

of

family 4,

Part

-

Il.

173-212.

and

geometry

associated

N.C.

Spernerraumen

Math.

Sem.

Ebene.

Abh.

of

designs

which

(1985).

Design

aus

einer

Univ.

vorge-

Hamburg

46,

25-54.

(1981).

D.

of

Springer-Verlag,

projective

Plann.

Arch.Math.

Hughes,

pp.

Statist.

(1977).

pp.

Mitt.

Reine

& Kageyama,

gebenen

Holz,

37,

Finite

(1978).

pp.

A.

16,

Wash.

Partitioning

Math.

designs. Herzer,

Dedicata

York.

(1985).

J.

Heft,

of

Geometries,

lismus.

III.

Planes,

Finite

D.G.

type

Proc.

P.(1968).

systems,

Glynn,

of

geometrical

design.

Pullman,

G.

D.

A

Press,

Can.J.

Glynn,

Geom.

355-361.

University

L.

G.

designs pp.

Conf.

berg—New

Ebert,

11,

balanced

(1973).

designs

Symmetric

H.

Intern.

Dembowski,

blocks.

1-24.

Inference

partially

J.

semi-translation

Symmetric

pp.

(1985).

Plann.

Cofman,

On

279-290.

Construction

37,

pp.

contain

a

unital.

179-183.

F. Piper, & R. University Press.

Cc.

Theory,

Cambridge

Barlotti: Finite geometries and designs

Kageyama,

S.

& Hedayat,

Statist.

Kelly,

G.

S.

A.

(1983).

Lecture

Kestenband,

N.

pp.

pure

with

M.J.

and

of t-designs

- Part

I1.J.

257-287.

Designs

Johnson,

in

family

7,

elations.

Kallaher,

applied

In

C.T.

"Finite

Long.,

mathematics

82,

eds.)

M.

Dekker

pp.263-272.

B.

C.

(1986).

elements.

Krier,

(N.L.

Notes

The

Inference

Symmetric

geometries" Inc.

(1983).

Plann.

J.

(1974).

Balanced

Statist.

Linear

incomplete

Plann.

block

Inference

representation

of

13,

derived

designs

pp.

on

q+l

45-50.

shear

planes.

Boll.

UMI (4) 9, pp. 709-720. Pellegrino,

G.

(1973).

4 3

e

su

Sulle

un

proprieta

BIB-disegno

della

ad

essa

11-calotta

collegato.

completa

Bollettino

di

UMI

(43°7, pp. 463-470. Pellegrino,

G.

(1974).

associazione

Costruzione

e di

di

una

PBIB-disegni.

classe

Atti

di

Accad.

schemi

Naz.

Modena

di

(6),

16, pp. 5-17. Pellegrino,

G.

(1977).

finiti.

Pellegrino,

Piper,

Piper,

F.

F.

G.

Schemi

Rend.

di

(1978).

in

a

25

Pps

plane

On

Math..15,

axial

(1983).

On "Finite

Long,

Raghavarao,

associated odd

negli

spazi

lineari

645-656.

order.

with J.

non

degenerate

Statist.

Plann.

conics

Inference

D.

(1971). of

D.K.

automorphism

of

symmetric

designs.

Ann.

S33—340).

pp.

designs which admit axial automorphisms geometries" (N.L. Johnson, M.J. Kallaher, C.T.

Dekker

design

for

pp.

symmetric

eds.),

M.

Ray-Chaudhuri,

of

9,

=sO0/—3ii2

(1982).

82,

associazione

(4)

t-designs

Galois

Disgrete

in

di

Mat.

Lectures

Notes

pp.

Inc.,

in

Constructions

experiments.

(1962).

constructing

pure

and

applied

mathematics

391-400.

J.

and

Wiley

Applications PBIB-designs.

combinatorial & Sons,

of

New

the

Ann.

problems

in

York.

geometry Math.

of

Stat.

quadrics 33,

pp.

1175-1186.

Ray-Chaudhuri, spaces

D.K. and

(1965).

Some

partially

configurations

balanced

incomplete

in

finite block

projective

designs.

Can.

11

Barlotti: Finite geometries and designs

J. Schulz,

Math.

R.-H.

R.-H.

Sperner,

B.

M.

114-123.

Uber

Zeit.

(1985).

sible

Segre,

pp.

(1967).

gruppe.

Schulz,

17,

On

designs.

Blockplane

98,

the

pp.

mit

classification

Europ.

J.

(1965).

Istituzioni Disegni.

Roma:

Istituto

Affine

Rdume

mit

(1960).

algebraische

Dilatations-—

of translation

Combinatorics

si,Reti,

E.

transitiver

60-82.

di

Geometria

Strukturen.

pp.

group-—divi-

369-374.

Superiore

III:

Comples-

Matematico.

schwacher J.

6,

Reine

Inzidenz Angew.

und

zugehorige

Math.

204,

pp.

205-215.

Vecchi,

I.

(1984). ovoids

10,

Some and

results

related

pp.219-225.

on

coverings

BIB-designs.

J.

of

Galois

Statist.

spaces

Plann.

with

Inference

.

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3

PORTRAIT

P.J.

OF A TYPICAL

SUM-FREE

SET

Cameron

School of Mathematical Queen Mary College University of London Mile End Road LONDON El 4NS

Sciences

INTRODUCTION A set

y € S, we

have

x+y

The

history

(homogeneous

relational

of

are

results

and

defining

among

dimension possess

tools

be

their

extremal

theory,

theory

model

and

that

I think be

won’t

built

a

of

used

find

the

"typical"

the

do

not

I

remarkable

methods

used

sum-free

set

these

until

on

our

we

areas

may

problems,

and

another’s

one

learn

in

are

Hausdorff

and

in

specialists

its

of

Firstly,

a

probability

effectively

out

life

a

on

such

until

section

first

reasons.

by

here

reported

takes

main

Secondly,

topology,

-_

could

we

from

(or

counter-intuitive;

and be

can

notion

the

them.

which

two

for

topic

surprising

areas

but

versa;

vice

analysing many

from

taken

date cannot

detail in the

quickly

understood.

better

is

case

special

all x,

vertex-transitive

to

planes,

in more

subject

theory

general

a

that

believe

to

They

investigations

the

to

the

quite

often

said

rise

projective

I describe

this

chosen

I have

own.

if, for

numbers

in Ramsey

applications

in

be

sets.

give

groups),

led

However,

paper.

the

can

natural

sum-free

many

which

applications,

these

sum-free

structures).

first

was

I

sets the

that

sets

blocking

theory,

graph

sum-free

yielding

graphs,

triangle-free

of

abelian

finite

in

is called

5S.

finitely

into

partitioned analogues

numbers

proved

Schur

when

1916,

S of natural

language.

A Many on

of the

the

basis

experimental

time

but

subsidiary conjectures of

stage.

reason were

computation.

Moreover,

little memory,

and

are

involves formulated Indeed,

typical

best

the

collection

(and

a few

the

subject

computations

performed

on

evidence.

of

of them is

require

still

proved) at

a lot of

a microcomputer.

an

Cameron: Portrait of a typical sum-free set

Readers

are

encouraged

to

try

their

hand

at

further

data-gathering!

1

ORIGINS

1.1

AND

Schur,

van

I have cannot

be

first

result

is

what

theorem

arbitrarily

(1936)

must

superior are

density

and

equal,

was

upper

(1927) sets,

the

This

density

led

numbers

exists,

years.

Ramsey:

must

Erdos

if

contain

and

Turan

of positive

upper

progressions.

of a subset

respectively,

limit

them

N the

(excluding

predates

of

long arithmetic

lower

theory

was

by fourteen

also

one

of natural

inferior,

later,

hard.

But

theory.

he in

was

proved 1977,

of

the

S of N are

|SN

the

{1,....n}|/n;

common

value

if

is

the

such

numbers

analogy

suggests

the

sum-free

density set have

1.2

the

progressions

general

enriched

(1952)

a

proved

arithmetic

The

nicer

that

a set

progressions.

by Szeméredi

conjecture.

gave

both

result

(1969);

proof

is

six

quite

proof

using

ergodic

and

ergodic

theory.

combinatorics

is possible

is obviously a problem

we

consider

set?

and

graphs

and

isomorphism

between

finite induced

extends

be

6.

has

sets.

The

density

detail:

In particular,

what

set

of

%

But

the

can

be

said

does

a "typical"

is it?

cyclic graphs.

finite or countable

is said

of [ which

in

if so, what

graph T

to

sum-free

and

consider

of a sum-free a density,

for

sum-free

A

automorphism

Roth

three-term

Furstenberg

Homogeneous We

contains

for discussion.

(1983)

No all odd

slow.

to four-term

His methods

See Petersen

initially

density

improved

years

graphs.

Ramsey

that

This

of S).

of positive

about

and

that

Progress

This

Ramsey

arbitrarily

limit

so

as

many

(1916)

sets.

pre-dating

Waerden

a set

theorem

sum-free

progressions.

that

contain

limit

der

and ‘Furstenberg

Schur’s

many

known

finitely

(The upper they

now

arithmetic

to conjecture

density

is

into

long

finitely

priciple),

of van

partitioned

Szeméredi

mentioned

into

pigeonhole

A similar N

in

Waerden,

already

partitioned

Dirichlet’s

APPLICATIONS

der

simple

homogeneous

subgraphs

Thus

undirected if, whenever

of [, there

"homogeneity

6 is an

is an

of [" means

Cameron: Portrait of a typical sum-free set

"transitivity

of the

non-edges,

Gardiner

etc." The

finite The

non-trivial

by

Henson

(1971)

(i)

homogeneous

graphs

(iii)

of m complete

m,

finite

or

countable;

(i)

(complete

n are

of

where

n,

the pentagon;

of K3 33

graph

(v)

for

each

n > 3, a graph Gp containing

and homogeneity

Types are

finite

=

(n

G,

automorphism

3)

graphs

Let

isomorphism:

Furthermore,

vertices

S = { nin x;

is

and Rényi

are

of [T as

joined

« up

to

Xj

x,

(iv)

in

the

interested

observed

cyclic

what

R and

that

graph

(n € Z) so that « maps

and

to conjugacy

Gg

not.

of a countable

to X,}. if

(iii) and

question:

the

Henson

automorphism

3 0, Xp joined

S determines

we

so

consider

have?

while

interest,

no

but Gy, (n ® 4) does

Let « be a cyclic

the

it);

graph R (Erdos

here;

We

R.

and

cyclic automorphisms,

number

of

(ii) are

relevant

do these

characterise

or "random"

(i) and not

(these

subgraphs

(1964)).

Rado

(1963),

having

no K, but

induced

as

(v)3

of

"universal"

the

(vii)

graphs

free

K, -

finite

complements

(vi)

Xn + 1°

The

graphs);

multipartite

line

can

and

earlier);

of size

graphs

the

graphs

by were

graphs

complete.

is

list

(iv)

being

determined

found

one

the

that

unions

properties

to

were

homogeneous

from

(apart

edges,

condition.

graphs

disjoint

all

We

strong

are

complements

(ii)

of [ on vertices,

countable

proved

(1980)

Woodrow

&

homogeneous

admit

group

this is a very

(1976).

discovered Lachlan

automorphism

Clearly

Then only

S determines if

in Aut([).

lj

-

il

(More

I.

xy, to [T up €

S.

Cameron: Portrait of a typical sum-free set

precisely,

if x, and

the associated

«3 are

subsets

cyclic

automorphisms

of N, then

«, and

&> are

of [, and conjugate

S, and

Sz

in Aut(I)

if

and only if S, = S>). The

I recall finite

that,

for

A subset

oO =

zeros

and

to R if and

includes

in

Clearly

no

sum-free

the

only

restriction.

elements

there

differ

k apart So

exist

ej-re4

(ii)

every

S is isomorphic of the

senses

universal.

exist,

of

G3,

if and

so

we

that

in

that

S, is sf-universal

if S

if k € S and

occur

say

begin

only

by k, and

can

we

it is not

too

to

than

i, j withl

the what

sum-free

Finite

What

«i

= 1 and j — i € 8;

© is a subsequence

is isomorphic

1.3

words,

(This

R admits

so

S.

a

finite

But

this

sum-free any

the

sum-free.

S is sum-free,

no

if, given

with

is

sequence should

set finite

S,

be with

zero-one

O = (e,,...,e,), either

(i)

typical

is

such

sequence

contains

by

in either

sets

be universal:

of S can

function

infinite

if S

set

any m

automorphisms).

set can

in positions

questions:

universal

is triangle-free

ones

trivial

2%o

exists

(In other

an

determined

typical

[(S)

with

Now

a

there

only

(1984);

if, given

if e; = 1.

Moreover,

automorphisms

no

characteristic

paper,

ones,

function,

[(S)

in Cameron

universal

and

if and

cyclic

then

sequence

graph

that

in detail

only

universal

the

cyclic

that

two

is

this

assertion

For

S if and

if S is universal.

non-conjugate

observation

S

Now

later

the

of zeros

its characteristic

ones,

only

described

m+i€

S with

sequence).

considered

of N is called

(€4 s0005@y))

1 ¢ i =n,

identify

finite

ao

[ = R was

results.

sequence

if we of

the

case

hard

Gg

analogous do

with

or

of S.

to show

if and

< Jj

vertices

Set o(i) = 1 if vo is joined to vj, 0 otherwise, for

Proof. i=

Let

3.4.

{Vo, V4} an

Vy}>

Hamiltonian

The cycle

result follows is obtained

on

by homogeneity.

by choosing

n = 2

edge.

al subset of Sf It is not difficult to show that a residu In particular, many density zero. consists of sequences with lower is not This zero. density lower have sequences sf-universal long uces in Proportion 3.3 introd surprising; the implicit construction are lacking: But results in the other direction blocks of zeros.

Cameron: Portrait of a typical sum-free set

28

Problem

(i)

Is it true zero

(ii)

Prove

An affirmative

more

interesting.

next

chapter

about

general

obvious

way.

members

of which

is true

of 3Pf).

of 2N

+1

of this

For

function

surely bounded

density

give

upper

of n,

bounds

Proposition

3.6.

density

at most

Proof.

By the

then

from

on

is not

zero.

provide

is much

typical.

this

The

section

has

However,

function the

0.

lower

In some

by Roth’s union

that

4;

it is an

e,(n)

density cases,

easy

as

this

and, all members

exercise

above

Sf,

sets

member

set

of members such

X, all

of the power

almost

is bounded

set

theorem,

a random

in the first

density

of the

in the most

a perfect

(disjoint)

checked

a version

generalise

(In fact,

the

surely %.

to which

to define

6/, of lying

expansion

away

space

of introduction,

It cannot

It is easily

have

if the

that,

also

of Schur’s

Sf as measure

extent

it is easy

consider

almost

show

of

By way

is the

density

this,

set

linear

example,

Also,

the second

and would

version

sf-universality

holds.

probability

on

(i),

Measure

to this.

Numbers

have

set has

any sf-universal

considerations.

and of 4N.

conditional

density

1.

question

Law of Large

settle

the structure

is devoted

that

of a "density"

In particular,

A natural Strong

lack

in Section

Generalities

some

in Sf with

zero.

(ii) would

the

By contrast,

contains

of sequences

the conjecture

density

to

for

discussed

3.3

has

answer

replacement

theorem,

the set

or refute

sequence

some

that

is residual?

of

to

by a

of X is almost one

can

also

density

A random

member

of Sf almost

surely

has

upper

yd

Strong

Law,

given

€ \ 0, at

least

1 - € of the

inputs

large

length

n have between

(4 - €)n and(% + €)n ones,

the first

which

occurs

not

position

with

later

than

€n.

But

an

input

this

of

of

29

Cameron: Portrait of a typical sum-free set

property bits

generates

of input

position

output

output

copied

m, then

x + m, where

the

an

are

at least

the

most

It

a

at least

output,

and

(9/2 -

if the

have

has

in position

x.

-

2€), and

the

to

prove

a one

(% + €)/(°/2

2€)n;

first

(% - €)n - m positions

output

is at

of length

to the

for

one

the

n

occurs

in

density

of

the form

Thus

the

limit

superior

is at

most 1/3. Odd

and

Fib

elaborate

The

following

result.

Theorem

4.1.

entirely

specific

...

method given

bounds

that

probability can

3[%n] terms.

This

« 33; it turns Py

to

is given

of proof

- P can

paragraph,

giving

consists

set

sum-free

upper

obvious

that

bound

of Sf satisfying Py

from

s(2i)

a

of the

closed

this condition.

smaller

of py;

= 0 for all i 2k, A; C S,1 2. By using a ramification

system

we obtained

(r—1)-times iterated exponential.

an upper

bound

for f (n;k,r)

as an

In fact we proved

f (nsksr)'™ < k®

ae

r-1.

Hajnal, Rado and I [XII] later showed that in fact f (n;k,r) is greater than an

(r—2)-times iterated exponential. exponential gives the correct bound.

It seems likely that the (r—1)-times iterated Let us restrict ourselves for the moment to

r = 3. The probability method gives without any difficulty

f (52,3) > 26"

(8)

and Hajnal proved more than 20 years ago that (see [XVIII]

f (n;4,3) > 2¢°2”

(9)

63

Erdés: My joint work with Richard Rado

Hajnal and I have a slightly better upper bound then (8) for f (m;3,3). Probably

f (n;2,2) > 2?"

(10)

but, unfortunately, on this no progress has been made.

This is one of the outstanding open problems of the subject.

I offer 500 pounds

If (10) holds then it would follow from our

for a proof or disproof of (10).

methods that f (n;k,r) increases as an (r—1)-times iterated exponential. Hajnal and I have a forthcoming new paper on f (n;k, 3) which will show that

in many ways f (n;k,3) behaves differently then f (m;k,2) which perhaps will further increase the interest in the fundamental conjecture (10). In [III] we give the first non-trivial lower bound for the van der Waerden

function W(n).

W(n) is the smallest integer for which if we divide the integers

an not exceeding W(n) into two classes then at least one of the classes contains

arithmetic

progression

of m terms.

The

only known

increases as fast as Ackermann’s function.

upper

bound

for W(n)

We easily showed by the probability

method that

W(n) > 2"? This was

improved

by Wolfgang

Schmidt

to W(n) > gito())n | Berlekamp

[3] showed that if n =p is a prime then W(p+1) > p2?. Lovdsz and I noticed c2"/n. that the local Lemma of Lovdsz gives, for every n, W(n) >

In some of my

As far as | papers I somewhat carelessly stated that in fact we get W(n) > c2".

64

Erdés: My joint work with Richard Rado

know it has never been proved (see e.g.,the survey paper of Graham which appears in the same volume as this paper.)

and Rodl

The first task would be to prove

W(n) > c2”

(11)

W(n)/2" — 0

(12)

and then to prove

(11) and (12) will perhaps not be difficult and I offer 25 pounds for a proof.

It

seems very likely that in fact W(n)!“ — , but perhaps the proof will require a significant new idea. For a long time all of us believed that W(n) certainly increases much slower

than Ackermann’s function. doubts about this.

W(n)

is much

The large majority still believes that the order of magnitude of

less than Ackermann’s

Paris and Harrington functions

which

As far as I know Solovay was the first who expressed

showed

increase

much

function.

The very surprising results of

that simple combinatorial

problems

faster than Ackermann’s

function.

can lead to Denote

by

f° (n;k,r) the smallest integer for which if we divide the r-tuples of the integers not

exceeding

f*(n;k,r)

into

k classes

then

there

always

is a

sequence

a, < a2 2r+1

and we also proved that every graph of chromatic

number 2X; contains a k (n;8} but does not have to contain a k &o, Xo) Unfortunately Walter cardinal

Taylor. m

all of us missed the beautiful and fundamental Let G be any graph of chromatic

there is a graph G,, of chromatic

number &:.

number

m

conjecture of

Then for every

for which

all finite

subgraphs of G,, are subgraphs of G too. Hajnal, Shelah and I have a triple paper on this subject where we prove some partial results and recently Hajnal and Komjath [10] have a paper in which they

prove many

further interesting results on finite and denumerable

graphs of chromatic number

subgraphs of

2 &:. In a triple paper of Hajnal, Szemerédi and

myself [11] we prove many interesting theorems and raise many problems which I

- hope will lead to further interesting results. Thus our old paper with Richard leads to many developments and I am sure will continue to do so.

In [XII], many results are proved and very many unsolved problems are posed

but their discussion on the one hand would lead too far into set theory and also a proper discussion of them would need a better knowledge of the many results on undecidable problems with which I am not so well acquainted.

recent Hajnal,

Shelah and many others could do a better job of this than I. Thus I will restrict

74

Erdos: My joint work with Richard Rado

myself to a very small sample.

Hajnal often observed that to prove positive results

in partition calculus we essentially use only two tools. and the canonization Lemma.

The ramification systems

In its most general form the canonization Lemma is

stated in XVIII p. 164 (see also [XII]). This Lemma was one of our most original contributions to set theory and it was very useful in many applications. Shelah has a very significant improvement of our Lemma

for r = 2 (see p. 159 of [XVIII]).

To avoid a complicated formalism we state our Lemma in only a special case: @ be a regular cardinal. of cardinals.

Let

Let Sg, 1 < 6 < wy, be a rapidly increasing sequence

Split the r-tuples of

(J Sg into fewer than w, classes (each r-tuple B 0 and

PG

F C |,| then for n > no(k, t), t

lAl < |,_,|

holds.

To see that the inequality (1) is best possible, ie., max|F¥| > [z7p consider

82

Frankl: The shifting technique in extremal set theory

the family consisting

all k-subsets

of X which

contain

¢ fixed elements.

For

n < (k—t+1) (t+1) a larger t-intersecting family was constructed in [F1] and [EKR]. (See next page). Denote by mg (k,t) the Jeast integer such that (1) holds. For n < 2k any two k-subsets

have nonempty

intersection,

that is A

is

intersecting; and it was shown in [EKR] that mo (kK, 1) = 2k. Hilton and Milner [HM] proved that for t = 1 and n > 2k the optimal family is unique.

In fact we have ng (k, t) = (k—t+1) (t+1) for all ¢ and k as it was proved for t > 15 in [F1], and for all t by Wilson [W].

Moreover for n > no (k, t) there is

only one optimal family. However, for t 2 2 one may ask, what is the maximum size of a f-intersecting

family

F,

FC

fl for 2k —t

Clearly, s/; is t-intersecting.

n=

One can also check that |.,| = |Ao| = a
15 and 0.8(k —t +1) +1) t, we may assume |4; N Ao| < |B, N Bo|. This implies FEB, NB, and fi, j} M.A; NMAp=. Say j ¢ Ay. Then A; =S;;(B,) = (B, — {j}) U fi). On the other hand A, = B,. Why did we not shift B, when i ¢ B» and j € By?

The

only

possible

|4; 9 A2|=|B,

reason

is

1 B3| >t8

B3 = (B,-— {j}) U {i} € F

Consequently,

It is not hard to see that if we keep on shifting then finally we end up with a stable or shifted family G, ie. S;;(G) = G for all 1 a

k-cascade 2>t DZ.

a;

a

fee é

(where

b

is

understood to be zero for b < 0). We leave the easy proof to the reader.

Theorem 3.2. |F| =m

(Kruskal-Katona

and

Then for all 2,1

3.1)

(3.2)

k

m=

cE +ese

theorem) a; +

t

Suppose

¥

is a family

is the k-cascade

of k-sets,

representation

of m.

|0e(Ak, m))| .

Because of the k-cascade representation, Theorem 3.2 is often clumsy for applications. Lovdsz proposed the following weaker, but handier version. Recall that

A

x(x —1)-..:G-a+t+1)

k is

i} Then for all 1 < @ < k one has

(3.3)

x hae

|, (F)| 2

Following [F2] we give a unified argument yielding both results.

Proof of Theorems 3.2 and 3.3. First we note that it is sufficient to settle the case £=1

(and

then

iterate

the

result

¢-times)

—

this

is, in fact, trivial

from

: i Proposition 3.1(ii) and the monotonicity of q| for x 2 a, respectively. Our next observation is that for all 1 < i < j one has

HS;(P) C S,;(P) —

a fact which can be proved by a simple but somewhat tedious case by case

analysis. Therefore, in proving (3.1) and (3.3) we may assume that F is stable (i.e., Sij(A) = F for all 1

not

true

|Fo| >

then

contradicting

—']

a feiand (3.4) would imply that if (3.6) were

:

In fact, |F| =

k=1

(3.5).

, so Therefore

that (3.6)

by induction is

true.

|@F| > Ree b¢ —

oF

CF, U {{I}

f-1-

note _ that

UG:G € 9A}. By induction |@F,| > |, _.], and thus

88

Frankl: The shifting technique in extremal set theory

aaa

|aF| >

r—-11

Xe

+

|e—21

x

=

;

[e—1)>

Proving (3.3).

To prove (3.1) we first show that one can assume that

LP plain

(3.7)

(|

a,—1

3

Porsche

lypal

ale

If this were not the case then (3.4) would imply that

(3.8)

a,—1 k

|Fo| >

foes

a;—1 t

+

+ bs

If a; — 1 > t, then we can forget about the +1 in (3.8) and deduce from the induction hypothesis that an

|AFo| >

1

ken

Henk

a,—1

Lathes

lapreay | , contradicting (3.5) .

If a, = ¢ then let s be the largest integer so that a, = s holds,

k

2s 2 t. Then

(3.8) can be rewritten as a,—1

| Fol >

A541) —1

Ey

PRG, Se

Ry

WE 5

From the induction hypothesis we infer that =

|9Fo| >

a,—1

a4,

Baie

es

a,—1

2

PTA

Pag)

S

PM

is—T

as4,—1

LSA i

ay—1 =

—1

s i

+ (+3)

a,—1 +

Crh

+

‘a

:

again in contradiction to (3.5). Therefore we can assume that (3.7) is true. We conclude the proof of (3.1) as that of (3.3), i.e., S laF| > |aF| a,|\F,|. By ar k

the induction hypothesis and (3.7), |@F,| > | beni

a; —

tbo”

|

t—2

|Adding

this inequality to (3.7), (3.1) follows. @

Proposition 3.4. Suppose that F is a family of k sets, |F| =m > 1 and x > kis defined by m =

x kI:

Suppose further that |d,(A)| =

x ce‘l holds for some @,

Frankl: The shifting technique in extremal set theory

Xo

< @

k-1|°

y>x,

then

ahsg|> 2 contradiction.

Theorem 3.3

implies

™ Let us mention

that

recently Furedi-Griggs [FG] and Mérs [M] characterized those triples (m, k, @) for which #@(k, m) is the unique optimal family in the Kruskal-Katona Theorem.

Corollary 3.5. (Sperner [S])

WANA Proof.

>

Suppose

that

@+*xFC

n n es |/ | Aholds with equality if and only if F =

Note that

x

X

|,.

Then

| Al

x

eo) (i=kiMk — 2x —k+@)+..-—k

+1)

is

monotone decreasing and apply Theorem 3.3 together with Proposition 3.4. @ Note that this corollary can be easily proved by a direct double-counting argument, too.

4. SHADOWS OF t-INTERSECTING FAMILIES Xx If one assumes that ¥ C

Katona

theorem

for

| iis t-intersecting then the bound of the Kruskal-

|, F| can

ld,F| > |Al for & < t. Let. us

first consider

beat = ie

o/ =

be improved.

[1,2k —¢]

| :

In particular,

| Clearly

we

shall

so is t-intersecting

show

and

. The next theorem shows that o/ is the “worst example”.

Theorem 4.1 (Katona [Ka2]) Suppose that ¥ is a k-uniform, t-intersecting family. Then forl 1+i holds. value of i for which this holds, ie. IF N[1,t+2j]| i(F) one has |F A [1,1+2i(F)]| =1+i(F).

define

for

[2¢+i+1,n]

ket-i

A€E

F

Fy ={F € F:i(F) =i,

[(2t+i+1,n] = A}. Then we have the partition F=

va

U

U

0

Define pF, ={G U A:G € 0,4}. It is immediate that 0%, C a¢Fz4 holds.

“We claim that for A, A' distinct 0,F%4 N Oe F4, = @. Suppose |A| =k —t —i, |A'|_ =k —t —i',i i — t + 1. Thus to have equality, one

Ae

must have f, = Ofori t

is

equivalent

Thus the Katona Theorem can be restated as follows.

Suppose that

F C 2* satisfies for all F, F' € F,

b |. i=0

and |Al


5

O0 2k there is a unique optimal family in the Erdos-Ko-Rado Theorem, as well.

6. THE HILTON-MILNER THEOREM. As we saw in the preceding section, the Katona Theorem implies the ErdésKo-Rado Theorem together with the uniqueness of the optimal families for n > 2k. Hilton and Milner described the next two optimal families. Define

A=

Clearly,

HO

G=iGE

[1,n]

+H E is

k

intersecting

and

we welU (ak,

NH = @.

Define

also

By : |G 9 [1,3]| > a Note that @ is intersecting, NG = @

and for k =2,

G=

Theorem 6.1.

(Hilton-Milner

intersecting,

6.1)

1eH. Dk+a

H# holds.

n > 2k and

Theorem

NF =.

([HMI))

Suppose

that

¥C

k

is

Then

-1 —k-1 IAI < [oA= i) - fre

Galas

Moreover, equality holds in (6.1) if and only if F is isomorphic to #, or k = 3

and F is isomorphic to G.

The original proof of this theorem

is rather involved.

For other proofs cf.

Mors [M] and Alon [A]. The present proof is due to [FF]. Proof. We start by applying the (i, j)-shift to ¥ Then either Si;(F) satisfies the

assumptions of the theorem or i is contained in every member of S;;(F). In the first case we keep on shifting until, eventually, we obtain a shifted family satisfying the assumptions.

Suppose now that at some point the second possibility occurs.

Without loss of

generality suppose i = 1, j = 2. Since 1 € F for all F € Si;(F), {1, 2} intersects

all members of ¥. Taking

(6.2)

Since

¥ of maximal size we may assume that

(6[12IcGe

few.

xX p

N ¥ = @, we may assume that {1,3,4,..,kK +1} €F

S'\2 we keep applying the (, j)-shift for 3 < i < j

O with IF N[1,ri + £]| > & —1)i + 4. Clearly, (7) >0. In the geometric

language, @ is the largest integer such that no walk w(F), F € F, lies entirely under the line y = (r — 1)x + @. The next proposition extends Proposition 7.2 in another way.

100

Frankl: The shifting technique in extremal set theory

Proposition 8.3. Suppose intersecting. Then

(8.5) Proof.

that

F,, ...,

F C 2* are

MF,) + +--+ +F,) Set dj =\(¥F;).

By stability for

stable

101

and

cross-wise

t-

2B rt holds. 1 [@ apr] holds

In view of Proposition 8.1 we can choose ¢ 2 0 such that

(8.7)

Dil zl— Fl ZI-Fl>° yp [@-rr] >e- Br. 1t+2s—3]A|.

o

Theorem 9.4. The following equality and inequality hold.

(9.3)

fm, 3,t) =2""

for i 2, n > 2s and

Then

- |

if and only if either xX ) :G

on

G =

rf

| | for some

for some Z €

xX Poet

Proof (Akiyama-Frankl [AF]). In proving (10.1) we may suppose again that @ is stable. Since Y contains subsets is not in G.

no s pairwise disjoint edges, one of the following s

G; = {i, 2 +1-i},

i=1,..,5.

However, if G; ¢ G, then the stability of G implies

xX

:-GN[1,i-1]#@

orG C [1,25 —i}}

Note that actually G; contains no s pairwise disjoint edges. from

gmax IG:l=19,|

Now

(10.1) follows

or |g,|

and equality holds only if

G=G,

or

G=@G,.

Finally, note that if @ contains no s pairwise disjoint edges and Sij(F) isomorphic to Gy for some 1 < ¢ < 5, then @ is isomorphic to Gy as well. @ The families corresponding to G, and @, for k-graphs with k > 3 are:

n= fre

xX

k :Fn

peel

ie-ile oland F,= |

is

Frankl: The shifting technique in extremal set theory

x K\>”

Conjecture 10.2 (Erdos [E]) Suppose that ¥ C

105

> ks and ¥ contains no

S pairwise disjoint sets. Then

(10.2)

|F| < max | 4~

—

" ih

1

=

ks — 1 k

Erdos [E] proved this conjecture for n > ng(k,s).

‘holds .

The bounds on no(k, s)

were improved by Bollobds, Daykin and Erdés [BDS] who showed that (10.2) holds for n > 2k3s. Firedi and the author (unpublished) proved (10.2) for

n > 100ks”, but to prove (10.2) in full generality appears to be a very difficult problem.

Let us prove an upper bound, which is not too far from (10.2) and holds for all n2ks. Theorem 10.3.

Suppose that

¥ C

xX k\> ” 2 ks and ¥ contains no s pairwise

disjoint edges. Then

-1 lIFl < (s-1) a

(10.3) Proof.

holds.

Note that for s = 2, (10.3) reduces to the Erdés-Ko-Rado Theorem.

fact, our proof will be similar to that.

First we prove (10.3) for n =ks.

ema O Gs Ul yer UNG. be an arbitrary IG;| = --- =|G,| =k. Out of these s sets at most Averaging over all partitions gives

eel]

In

Let

partition with s—1 can be in F.

Now we apply induction on 71 and prove the statement simultaneously for all k with ks < n.

Again, we may assume that ¥ is stable. Consider F(n) = {F € ¥:n ¢ F} and An) = {F— {n}:n € F € F. We claim that neither of them contains s pairwise disjoint sets. Indeed, this is trivial for F(n) C F. As to F(n), note that if H,,...,H, € Alm)

are pee

disjoint then choosing s distinct elements

n—s(k—-1) 25, DP avycetsse “from [1.n]—(H, U --- U A,), which has size the stability of F implies F; = (H; U aN € ¥. However, Fj, ..., F, are pairwise disjoint, a contradiction. Now using the induction hypothesis we infer that

)| < oo) + |F)| |F| =|Fa

als

1)

Frankl: The shifting technique in extremal set theory

106

11. ON r-WISE INTERSECTING FAMILIES Recall that

F C 2% is called r-wise intersecting if

for all Fy,...,F,€F F,

++:

F, N --:

N F, # @ holds

If |F,| + --- +|F,| > @—1)n,

then necessarily

1 F, # @ holds.

This shows that the assumptions of the next result

are necessary.

Theorem 11.1

([F8])

rk < (r —1)n.

Then

Suppose

that

xX k

FC

is

r-wise

intersecting,

-1 IF < lea

(11.1) Moreover,

F={FE

excepting

the case r= 2, n = 2k equality

holds

if and

only if

be € X} holds for some x € X.

Neither the original nor the present proof uses shifting. However, the present proof uses the Kruskal-Katona Theorem, which we proved by shifting. First we prove a proposition which is due to Kleitman. Proposition

11.2

(IKL)

Suppose

_ that

xX

FC

A;

Fe dines

T

Nara cine ate al so

F, U-+::

If

(11.2) Moreover,

X=G,U

are

no

n

IlFil/|,| ,..., A, are events in a probability space having dependence graph I’, and there exist positive y;, y2,..., y, satisfying

logy;

>

>

yjPr(A;) + y,Pr(A,)

jer

for 1

ck?/(log k)? for a suitable c > 0. The following result gives the sharpest known bound currently known.

Theorem 2.4 (Sp77]

(2.3)

r(k, 3) 2

ees (1) |x0og k)?. 7

Proof: The proof is a modification of that of Spencer [Sp77]. Let the edges of K, be independently 2-colored red and blue with the probability that an edge is colored red always being p. To each 3element subset of vertices S associate the event As that all the edges spanned by S have been colored red. Similarly, to each k-element subset K associate the event Bx that all the edges

spanned by K have been colored blue. Observe that

r(k,3) >n if Pr(Q) As N (1) Bx) > 0. Ss

K

Let I’ denote the graph with [i] ar (7] vertices corresponding to all possible As and Bx, where

{As, Bx} is an edge of I if and only if |SNK| > 2 (i.e., the events Ay and Bx are dependent), and the same applies to pairs of the form {As, As} and {Bx, Bx}. Let N44 denote the number of vertices

of the form As for some S joined to some other vertex of this form (so that Naa = 3(n — 2)), and let Nag, Ng4 and Ngg be defined analogously. In this case, Corollary 2.3 implies:

If there exist positive p, y, z such that:

113

Graham & R6édl: Numbers in Ramsey theory

114

p yPr(As)(Na4 + 1) + zPr(Bx) Nap , log z > yPr(As)Npga + zPr (Bx) (Nop + 1)

then r(k,3) >n. Now, k

Pr(As) =

p>, Pr(Bx) = (1 — p) f) < exp |-o(8]

Also, we have the bounds

n < ne Nan< (f} [2]met < neel, k

k

Naa+1 0 and k sufficiently large. 2.2 Constructive lower bounds

In the preceding sections, all of the bounds given were based on the use of the probability method. As a consequence, the proofs do not produce any explicit colorings but rather, they only prove that such colorings exist. To remedy this not entirely satisfactory state of affairs, attempts have been made over the years to construct good colorings, unfortunately without much success. For

the case of r(k, 3), Erdés [E66] has given a construction which shows that rk, a)"> Kato)

where

Sawer log 2 af ar metas | 13139 eels improving an breakthrough Frankl [F77], this section,

earlier construction of his which gave a somewhat weaker result (cf. [E57]). A for r(k,k) finally occurred several years ago, however, with the result of who gave the first Ramsey construction which grew faster than any polynomial. In we will outline a more recent theorem of Frankl and Wilson [FW81] on set

intersections which yields the best constructive bound for r(k, k) currently available.

Theorem 2.7 [FW81] Suppose Fis a family of k-sets of {1, 2,...,m} such that for some prime power q,

115

Graham & R6édl: Numbers in Ramsey theory

F,F'€ FEF #F'

116

> |FNF'| #k (modq).

Then

(2.10) Proof:

lA < (,"4}Let A;, Az... Ain) be all the j-element subsets and B;, B>,...,B (")be all the i-element Jj

I

subsets of {1,2,...,n}, where i < j. Define the ("| by ("] matrix N(@, j) as follows:

(u, v)-entry of N(ji,j) is 1 if B, C A,, and Oif B, ¢ A,, forl 0. Further, f satisfies

fd) + d* -— d)f'(d) . Therefore,

1+ (mi -—d,-Df@)

21+ M-d,-

21+(M—d,-—

Df)

+ @—-d,- 1)’ - a)f'(d)

If (ad) + dd, +d — 2d,d,)f'(d)

2 (n — d, — 1)f (da) + (a, + 1)f d)

by (2.18)

= nf (d) .

Thus, a 2 nf (d) as required, and the theorem is proved. @ As an immediate consequence, we have: Theorem 2.10

(2.19)

r(k, 3) < k?/log (k/e) .

Proof. Let G be a triangle-free graph on n vertices and suppose a(G) < k. Since the neighbors of any vertex v form an independent set then we must have degree (v) < k. Thus, the average degree

d in G is at most kK. Therefore, by (2.16)

(2.20)

k > alG) 2 nf d) 2 nf (k).

Hence, if n > k/f(k) then G must contain an independent set of size k + 1. This implies

(2.21)

Rk + 1,3)

n/(100d log d) for any triangle-free graph G with n vertices and average degree d. If 3 is replaced by an arbitrary but fixed value ?, then the best bound on r(k, @) is given by the following result of Ajtai, Komlds and Szemerédi:

Theorem 2.11 [AKS80]

(2.22)

r(k, £) < (5000)%k*'/(log k)*-?

for k sufficiently large (depending on ¢).

Graham & R6dl: Numbers in Ramsey theory

120

It will be convenient for the next result to define the related quantity r°(k, @), the largest value of n such that there is a red-blue coloring of the edges of K, having no red K;, and no blue Ky. Thus,

rik O=r(kK+1,2@+0)—-1. For arbitrary k and @ (not satisfying k >> ¢ required by Theorem 2.11), the best current upper bound is given by a recent result of the second author:

Theorem 2.12 [R]

(2.23)

rk, O 0.

Because of space limitations, we will not give the proof of (2.23). method used in proving it by proving the following weaker result.

Rather, we will illustrate the

Theorem 2.14

(2.25)

r(k, 2) @+a-p [7]-ea-p (2). Let d,, dz, ...,dy be the degrees of the vertices of G. Then

7G) + 7G) = [5-5 Bea, - vay - (5) - 3Be

Da - ap

> + (n(n — 1)(n — 2) — 36n(n — 1)?+36°n(n — 1)2)

- @

+0 -—) [%]-sa-a) [R}. =

Graham & Rédl: Numbers in Ramsey theory

121

Lemma 2.16 Suppose m and n satisfy m > e2n > 0. Then

(2.27)

r°(m,n) EA" s(s — 1)

te

s-1

jem,

1+ «| Pe jromn). sj s-—1

Proof: Set N: = r*(m, n) and let Ky be 2-colored so that no red K+, and no blue K,4; is formed. Let d,” denote the number of red edges incident to the ith vertex of Ky, with d,” denoting the analogous quantity for the blue edges.

If d;” > = 1+ + N for some i, then by considering the

“red” neighbors of this vertex we get (i). Similarly, if d,{“) > e 1+ feN then (ii) must hold. Hence, we can assume that

d?
B= Hli+E|[r+—ti}y

we

now

indicate

how

to

do.

First,

observe

that

by

(2652)

N ted @= - 26()]> 22m_ - ent S}-y so

that

(2.35) Thus

B >

it will

(2.36) |B o

be

enough

to

show

that

egen + © - 2 - 1)/ (i-1)

>

sept eeeler ar

for N> 16s? We will not carry out all the details of this computation, relatively straight forward (but unenlightening).The basic that the important term of (2.36) is

(2.37)

[2 r 8 s

which

can

(2.38)

be

y mec)

“sg?

In

turn

the

(2.39)

by the The

main

e7]

s(s-1)

definition

value

assigned

of to

s-1

€ a cet

¥ 1.) (< ms =i

n? st

m(m-1) }} s*(s-1)2

contribution

m2 —m(n-1)

s*

s

as

m2 m(m-1) iS = gla-t)

+

are is

1+ £|ie+ =

s(s-1)

rewritten

which point

_*

to

(2.38)

ut) a(n-1)]

turns

out

to

be

an

87(s-1)

s. € now

guarantees

(2.36), and

therefore

(2.34), holds.

124

Graham & R6dl: Numbers in Ramsey theory

The

case

that

way (using the to yield (iv). We

need

there

are

B blue

trinagles-:follows

in

exactly

symmetry of 8 and 1 - 8 in the expressions This completes the proof of Lemma 2.17.

one final

observation

before

proceeding

to

the

proof

Fact.

= he aS : i=r+1

(2.40)

Proof:

The

n

n 2 1/i

Siok i=

>log

m=k,

+ 1)

r WG

=log(s

Set

log(n

Consequently

s

HAN i=r+1

Proof of Theorem We will consider

-

“

i=i

increasing.

s

1.

s + 2 ie Gweall

sequence

a_=

is monotone

Sy wilioyee

coe oh wael =

+ 1) + es

Siiet naar

log(r

t=k+2&

*

i]

a

al EL

2.14. Let G be a graph with the following algorithm.

n=,

+ 1) -

and

log t Co|log

log t

ang

F

the

same

involved)

r*(k, 2) vertices.

of

(2.25).

Graham & Rédl: Numbers in Ramsey theory

2.

3.

If max{m/n, n/m} > x then halt; otherwise go to 3. Ifr‘G@n, n) < 16(m +n)? then halt; otherwise go to 4.

4.

Select a pair (m*,n*)

m'+n*=m+n—-—1 m=m’',n

for which

or

125

one of the possibilities of Lemma

m+n—2.)

2.17 occurs.

(Thus,

Let G* be a graph with r°(m'‘, n°) vertices.

=n’, G =G‘ and go to 2.

Set

Suppose now that the algorithm halts at some graph G’ of size r* (k', é’). Let t': = k'+ @' and

p=t-—t'. of e=

Note that if y: = dog lent then at each pass through step 4 of the algorithm, the value fon

min(m, n) 8Gn + a)

satisfies

ea (2.41)

Bi

:

8(m +n)

> 8(1 + m/n) 1 > ———

80x)

2

=).

Hence, by the time we have reached G’, we have (by Lemma 2.17) accumulated the “gain factors” to obtain the estimate

epee

ele

(2.42)

py

ke

Ue

aoa Tiel Ta

+ 2)

GAG

(EE

IS

+ 2). (ser)

Gee

y

Derr rr

2

Lee

ke

r°(k, @)

= a! 1+ oe r(k, @).

We now consider several cases, depending on how soon the algorithm halts, and why.

Case 1. t'> Vt +2. Subcase (a). r°(k', 2.) < 16 t ?. Since ¢ is large then so is t’. Thus, by (2.7) we must have min(k', 2’) = 2, say (by symmetry) é'=2.

Thus,

r(k', 2) =r°(k', 2) < (k'+ 1)*/logtk'+ 1)/e) (2.43)

< (k' + 1)?fog(Vt /e) = 2(k' + 1)?flog(t/e”) .

Therefore, by (2.42)

.

rk,

be

eee

t e< (+3) r°(k',£') 2

(2.44)

fort = k + @ large.

li) < —— “4 nip

. 2(k + 1)?

log (t /e?)

s|k+é | |og(k + £)

Graham & R6dl: Numbers in Ramsey theory

1 ee logt Ip! > x Sor Leer ORE Subcase (b). k'/£'

126

ie

Thus, by Lemma 2.16,

re’, e) (26 + p — 48"!

(where m-+> (k) indicates that m —>

(k) does not hold).

This together with Theorem 4.2

implies Theorem 4.4 For p 2 3,

(4.5)

Logy-2(rp(é)) > cp'é? .

The asymptotic behavior of r,(¢) is not known for p 2 3. However, because of the SteppingUp Lemma, any improvement on the lower bound of r3(¢) would yield a corresponding improvement on the lower bounds for r,(¢), p > 3. The major open problem here (and indeed, one of the main unsolved problems in Ramsey theory) is the determination of the order of growth of r3(é). P. Erdés is currently offering $500 for an answer to the following problem. Problem 4.5.

Is there an absolute constant

(4.6)

c > 0 such that

loglogr3(¢) > cé ?

It is interesting to note that if four colors are allowed (rather than two), then the analogue to (4.6) is valid, i.e.,

(4.7)

loglogr3(¢,4) > cé.

This is a consequence of the next result.

Theorem 4.5 (Hajnal; cf. [EHMR84], Th. 26.3)

If n-+> (2)3 then 2"-+> (¢ + 12. Proof: Let [n]?=C, U C2 be a 2-coloring with no monochromatic ¢-set, and let < be the lexicographic order on An), the power set of [n], given by: a, < az iff max(a; — (a, Nay) < max(az — (a, Na,)) . For a, ¥ a2 € An), we also define

8(a,,a2) = max{i: i € (a; — a2) U (a, —a))}.

For a1,a2,a;

€ An) with a; < az < a;, let 6; = 6(a,,a) ’ 62 = 5(a,a3)

Finally, set

.

Graham & R6édl: Numbers in Ramsey theory

{a),a>,a3}

E Sy iff {5,55}

136

€ Ci and 6;
62, {a},a,a3}

€ S3 iff {5,59} € Cz and 6; < 6,

{a},a2,a3}

€ S4 iff {5,55} € C2 and 6; >

6.

Suppose now that there is a family ¥ € An), |X| = ¢ + 1, which is monochromatic. [X}? C S, (the other three cases are similar). Write X= and let 6;: a

5(a;,a;+1),

{a),a>, wien e +4ea1},

1 < i < £.

Fori

ay,
S(é,m +1). For a fixed r, let M'=N(é,1,7™) and suppose x: [MM"]—>TIr] is given. Define the

x': [M'] —>

I[r™] s0 that x(k) =x'(k')

By the induction

x €[0, 2-1].

M=N(Z, m,r), induced coloring

hypothesis,

if x(kKM — j) =x(k'M -j) there

exist a’ and d’ such

for

O< jf 1 +> S(€+1,1).

For a fixed r, let x: [2N(¢,r,7r)] —>

Ir] be

arbitrarily given. Thus, there exist a, d,,...,d, such that for x; € [0, 2], a + > x;d; is bounded r

above by N(¢,r,r) and x(a + > x;d;) is constant on ¢-equivalence classes. i=l

i=l By the pigeon-hole

principle there exist u, v € [0, 7] with u < v such that

x(a + > fd;) =x(a + > d;) . i=l i=l Therefore,

x(a + Sed +10 SD d)) i=t

imutl

is constant for ¢ € [0, 2]. This proves S(¢ + 1, 1). Since S(1, 1) clearly is true then the theorem holds by induction.

&

Of course, Theorem 6.1 is the special case m = 1 in Theorem 6.2.

The upper bound on W(k,r) resulting from this proof is quite large.

Essentially, it is given

inductively by a function in two variables, and grows like the Ackermann function (cf. [Sp83]). In fact, no proof is known which yields an upper bound on W(k, r) which is even primitive recursive!

The same also applies to the special case W(k): = W(k, 2).

On the other hand, the strongest lower bounds for W(k) are much more modest, namely, just exponential in k (cf. Theorems 6.3, 6.4). What the truth really is here represents a central open question in this whole area.

Theorem 6.3 [Ber68] If p is prime, then

(6.1) Proof:

Wep+l1)>p-2. For simplicity, we only prove a slightly weaker result:

(6.2)

W(p +1) > p(2? - 1).

Let GF (2?) denote the finite field with 2? elements, and fix a primitive element a € GF (27). Let V1, +++» Vp bea basis for GF (2?) over GF (2). For any integer j, set a

=

41jV1 + arjvo t+. oe + apjVp

»

ay € GF (2) ‘

Graham & Rédl: Numbers in Ramsey theory

143

Let

Com (ji: aj =0,

1 x; (2d + 1)', 0 < x; < d, which are all at most n. i

There are essentially n2~* such integers.

Thus, for some s

n v3(n) > |Xn.a.s| > Fak Setting d = exp(vViog n) we deduce \Xn,d,s| 2 1 exp(—c Vlog n) for some c > 0, as required.

Instead of the upper bound in (6.8), we only show here that v3(n) = o(n). There are other relatively simple proofs of this fact (cf. [RS78], [G81]). The proof give here, based on ideas of Ruzsa and Szemerédi, is taken from [EFR86]. Theorem 6.7

(6.9) Proof:

v3(n) =o(n). Let G = (V, E) denote a graph and let A, B C V be a pair of disjoint non-empty subsets

of V. The density of the pair (A, B) is defined to be the ratio

d(A, B): = e(A, B)/|A||B| where e(A, B) denotes the number of edges {a, b} with a € A, b €B. The pair (A, B) is called e-uniform if for all A' © A, B' © B with |A'| > «|A|, |B'| > «|B| we have

|d(4', B') —d(A, B)| ,A,;#’,v)-Family.

Since

there

+

this

is a

cardinal

contradicts

# cf,

< \} and

it

follows

almost

2"

P is directed

is

a

disjoint axioms

we give

if there

1.2}

false,

there

is an elewent pair

2” + @_.

is an almost

question

theory.

introduction

be a partially

if every

that

natural

set

with

and

SETS

{#1,@,a)-family.

of

a brief

is consistent

DISOINT

(Corollary

that both the Sxistence,

ordered

r € P such of elements

to

This

To discuss to Martin's

set.

that

Two

ask

disjoint

is

turns

whether

out

this,

to be

and

axiom.

elements

P,q

Psranqir.

in P are

some

compatible.

eP

Whe The

Milner & Prikry: Almost disjoint sets partial

order

elements

is

a

ccc

if

on

the

a set

MA(x)

is

a

It

is easily

family

However,

“countable

size

of

MA(w)

is

axioms

of

ZF,

it

However,

it

the

order

to

appiy

principle

with

In fact

it

not

of fewer Clearly

does

is equivalent

CH

fundamental

as

MA,

an

result

A

and

subset

FnG

? is a CCC

partially

of

there

?,

then

D

but

it

¢ P

is

subsets of

¥ @ for every ordered

an

is

(G can be chosen

quite

same

set

F €

and

¥

set.

f-generic

to be an w-chain).

so MA(k) implies x ¢

naturally

turns

to

algebra,

the

many

the

of SOLOVAY

dense

out

CH,

appeal

the

to be

theory

sets

of of

other

forcing.

forcing

in

combinatorial

and

set

point

open

a compact

sets

(see e.g.

is

KUNEN

of

MA

is

which

is

important

% TENNENBAUM

the

a poverful

assertion:

disjoint

as

context

the

analysis

usefulness to

in

with

topological

nowhere

alternative

intuitive

familiar

which

to

but

the

to be

uncountably

2° closed

implies

employed

have

arise

axiom,

containing

incompatible

is true for every x, w < « ¢ 2°,

applications

than

if G is directed

is true

not

necessary

the

of

condition",

statement

does

is not

sets).

(see e.g. KUNEN [1980],p.54),

MA: MA(x) axiom

set

chain

If F is a family of cofinal

subsets

axiom

this

< d.

¢ x cofinal

that

unebuntable

incompatible

whenever

seen

Martin’s

for

assertion:

MA(2°) is false

Although

be

nod

stands

G C P F-generic

the

of

is

Space

P Banta lee

in P if (vpeP)(SdeD)p

P, we call

rs

is

historically

condition

cofinal a

?

(CCC

161

that

[1971]:

it

It

not

topology.

Hausdorff the

union

[1980],p. can in

65).

frequently view

of

is consistent

the

that

MA holds and 2”)0,. Many

For

example

TENNENBAUN

outstanding

MA(#,)

[1971],

problems

can

implies

the

easily

see also KUNEN

“more concerned

with

the

in particular

with

the way MA

subsets

of x (( a) by subsets

in MARTIN & SOLOVAY immediate

interplay

consequence

[1970] of

the

[1980], between

is used

be

settled

Suslin

p74}. MA and

of w (a technique

following

important

the

hypothesis

However, almost

(see Theorem

and JENSEN & SOLOVAY

with

aid

of

MA.

(SOLOVAY

&

here we shall be

disjoint

2.4 below)

families

,

to encode

due to Solovay described [1970]}.

This

will

be an

result.

If A, B ¢ P(w) are such that |4u2| < « and Assume MA(k). THEOREM 2.1. JB \ ua’| = @ whenever B € B and A’ is a finite subset of A, then there is set a

€ &.

C € w such that

CoA is finite and CoB is infinite

for all AE

A and B

Milner & Prikry: Almost disjoint sets

PROOF.

Let

=

(P,)

be

the

partial

162

order

on

the

set

P of

(s,F), where s€ [a], F € [a] and (s1,F,) < (S2,F2) F, and UFjns, in P if and

© 5,.

only

Note

that

(sus’,fuF’)

subset

which

first

the

same

For A € A,

let

Ban = {(5,F):

each which

has

holds

with

a

=

that

CnB

is

immediately is,

is

(5’,F’}

€ 5,

compatible

and

the

MA

are

with

for

all

€ G

is

n € a,

that each D, and

sets.

For,if

is

compatible.

a directed

these

A

compatible

is finite.

(s,F’)

B € & and

to check

there

F}.

so AnC

therefore

and

is easy

(s,F) and

set,

The

theorem

€ A,

there

with

(s,F),

If BE

&,

G,

is

then,

it for

and so CoB \ n 2? snB\n # @, and

oO

2.1

More

2.2.

are

? is clearly CCC since

two elements

AEF},

some

€ GE, has

mwany

question

consistent

(#,,,0)-family. THEOREM

any

is infinite.

ansvers it

are

by MA(x)

for

(s,F)

Theorem

that

(s’,F’})

extension.

It

so

intersection

€ UFns’

any n € w, there therefore

51 © S25 Fi ©

€ gs’,

and which

(s,F)€G

since

Ans’

UF'’ns

Da = {(s,F):

\ n # 9}.

U{s:

(s,F) € GoD, and, follows

component,

in P and

non-enpty

C

and

of ? there

let

soB

aoe is cofinal

Cs

is a common

in any uncountable have

{s,F),

pairs

if UFns'

in which case

two elements

all

applications.

raised

that

at

there

the

is

For

beginning

no

example,

of

this

it

section

maximal

almmst

disjoint

maximal

almost

disjoint

generally,

implies

that

there

is

no

(x,w,w)-family for w < x ¢ 2", PROOF.

Given

an

alwost

{with B = {w}), there

AEA.

is an infinite

(x,w,w)-family

,A,

set C C w almost

by

Theorem

disjoint

2.1

from each

o We

also

disjoint

should

consistent

Waximeal

mention

that:

almost

for

disjoint

that,

each

in

contrast

cardinal

(x,w,w)-family

x

-

to

Theorem

(w,< x

see

the

¢

2.2,

2”)

rewark

it

there

is

is

following

a

3.6

below. We

subsets

Obviously, length THEOREM

shall

A,B of there

w,. 2.3.

o,

use

write is no

However, MA

the

APB

following

consequence

if A \ B is

strictly

decreasing

infinite sequence

of

Theorem

and of

2.1.

For

B \ A is finite. subsets

of

w

of

... implies

subsets of w of length 2°.

that

there

is

a

bh-decreasing

sequence

of

Milner & Prikry: Almost disjoint sets PROOF.

Let

that

as,

Aa}

and

a

¢ 2” and

holds

B =

for

{Ag

a.

B

*

~

" a

'

a

‘

-

‘

4

’

-

— a)

=

a*

ie

ry

=

.

cae 7 »

-_

i

-

> = -

a

aaah

> >

-

the

’

-

-

THE METRIC STRUCTURE

OF GRAPHS: Theory and Applications

Peter Winkler Emory University, Atlanta, Georgia 30322 U.S.A.

Abstract. The "path metric" on the vertices of a connected graph G is given by defining the distance between two vertices u and v to be the minimum over all paths P from u to v of the number of edges in P. Over the past 20 years a structure theory has emerged whose

aim is to simplify the path metric by means of isometric embedding in cartesian products of graphs. Although the theory is peculiar to graphs, it looks like a typical algebraic structure theory and the analog of representation by subdirect products works out nicely. Development of the theory was motivated by a problem from computer science and has contributed to the solution of several others. These problems, which arise in complexity theory, network design and data structures, will be described briefly and their connections with the structure theory explained.

Introduction

The general aim of a structure theory is to break down a complex object into simpler components, the object in this case being the path metric on a graph. The approach taken is a familiar one to algebraists, namely representation by embedding

in a product; it turns out that this

approach works quite well here, despite the apparent absence of category-theoretic underpinnings. There are a variety of applications of the structure theory to computer science, some of which

will be briefly presented below. By a graph G = we shall mean a finite set V(G) of vertices together with a

collection E(G) of unordered pairs of vertices called edges, that is, a finite, simple graph. For definition of basic terms such as "path" or “connected” the reader is referred to any elementary text on graph theory or combinatorics.

The distance dg(u,v) between two vertices of a con-

nected graph G is defined to be the number of edges in a shortest path from u to v; the resulting "path metric” turns V(G) into a metric space. is a map a: V(G)—V(H)

An isometric embedding of G in another graph H

such that dy (o(u),a(v)) = dg(u,v) for all vertices u and v of G; the

image of G is then a subgraph of H isomorphic to G.

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Winkler: The metric structure of graphs

The Cartesian Product

There are at least three natural graph products definable on the cross product of the vertex sets of two graphs; a clever notation scheme due to J. Nesetrtil assigns a symbol to each which looks like the corresponding product of two edges. Although the "weak product” (denoted by x)

and the "strong product" (@) are universal products in the category of graphs (as non-reflexive and reflexive relational structures, respectively), the "cartesian" product (C1) has turned out to be extremely versatile and is the product of choice in metric theory, for reasons which will become plain. The cartesian product GOH

of graphs G and H is defined as follows. The vertices of

GOH

are ordered pairs (g,h) with geV(G)

GOH

iff either {g,g’} is an edge of G and h=h’, or {h,h'} is an edge ofH and g=g’. It

is

easily

seen

that

the

and heV(H), and {(g,h),(g’,h’)} is an edge of

cartesian

product

{(1,82 --- »8k)(81.82)---»8e’)} is an edge of GAGA

is

associative

and _ that

-- - OG, just when {g;,g;"} is an

edge of G; for some i and g;=g;’ for all 7, in which case we say that the edge "belongs to the ith factor." Note that the product of k copies of K2, which we denote by K§, is just the "Boolean k-cube,” i.e. the graph given by the vertices and edges of a (unit) cube in k-space; its metric is of course the k-dimensional Hamming metric. The cartesian product of K3 and the three-point path P; is illustrated in Fig. 1 below. Sabidussi (1960), and later Vizing (1963), showed that the cartesian product enjoys unique

prime factorization for connected graphs.

The proofs were difficult and did not provide an

efficient method for finding the prime factorization---a point which we will address again later. Unique prime factorization fails for disconnected graphs, for which Zaretskii (1965) gave the fol-

lowing example: K}+K7+K}+K2+K>+K, = (K3+K,0O(K3+K2+K) = (K2+K))O(K}+K}+K;)), where "+" denotes disjoint union and all the graphs in parentheses can be shown to be prime. Since each step of a path in a product []G; moves in only one factor, any path in a pro-

duct breaks down into paths in the factors. Thus we have that

Winkler: The metric structure of graphs

199

Oo

eee

©)

3 fe)=4

Fig. |

Winkler: The metric structure of graphs

Are (Sa + - Be)(81's- - - 8k’) = Lido,(81081) It follows that if G is a product of simpler graphs, or if it is an isometric subraph of such a product, then its metric can be broken down into simpler components.

Isometric Embedding in Products

In representing

the metric on a graph G, the best result one can reasonably hope to

encounter is an isometric embedding of G in K*; equivalently, a description of G as a subspace

of Hamming space. Such embeddings were sought by Firsov (1965) for the purpose of investigating "the closeness of various linguistic objects," but only preliminary results were obtained at that

time. In the early ’70’s Graham & Pollak (1971, 1972) highlighted the problem in the process of designing addressing schemes for computer communications networks. Information is sent from

computers in "packets" only microseconds long, and efficiency requires that many of these be on a network simultaneously. Each packet is augmented with the address of its destination, and must somehow find a short route to that destination through the network.

Graham and Pollak’s idea was to represent the network by a graph and provide addresses for the vertices in such a way that the distance between two vertices (number of edges in a shortest path) can be easily recovered by comparing their addresses. Then when a packet reaches a

vertex it can compare its destination address with the local address, discovering, ~~ that it is presently at distance k from its destination. It then checks the addresses of neighboring vertices until it finds one at distance only k—1 from the destination, and moves there. Thus using local information only, it finds a shortest path to its destination.

The most obvious way to realize Graham and Pollak’s objective is to address the vertices by binary strings whose Hamming distances match the graph distances; that is, to embed the

graph isometrically in a Boolean cube. Unfortunately, as noted already by Firsov, there are graphs which are resistant to such embeddings even when made bipartite by edge-subdivision. Graham

200

Winkler: The metric structure of graphs and Pollak then turned to the addition of a "don’t care" symbol c"*") to the binary strings, whose distances to the other symbols 1 and 0 are both defined to be zero. Even though the space of all such strings (of a given length) is no longer a metric space, the new scheme worked beautifully;

later Winkler (1983) was able to show that any graph on 7 vertices could be addressed by strings

of this sort having length at most n—1. The question of which graphs could be addressed by the {0,1} scheme, however, remained open until Djokovic (1973) gave a complete characterization. If {x,y} is an edge of a connected graph G, let Nxy = {veV(G): dg(v,.x) = dg(v,y}-1}, ie. the set of vertices nearer to x than to y, and xNy = {ve V(G): dg(v,x) = dg(v,y)}. Further, any set S of vertices of G is deemed convex if every shortest path in G between vertices of § is contained completely in the subgraph

induced by S.

Theorem (Djokovic 1973). A connected graph G can be isometrically embedded in a Boolean cube if and only if G is bipartite and for every edge {x,y} of G, both Nxy and Nyx are convex.

We omit the proof of Djokovic’s theorem since it will follow from more general results below.

If more symbols are allowed in the addressing scheme, with the distance between distinct symbols still always 1, then the problem becomes one of embedding G isometrically in some power K* of a complete graph. This also cannot generally be done, as can be seen for example

by examining the complete bipartite graph K2, or the cycle Cs. A polynomial-time characterization of the graphs which can be embedded in a power of a complete graph (equivalently, in a product of complete graphs) was obtained by Winkler (1984), who also made the (then) surprising

observation that all irredundant isometric embeddings of this sort were unique up to symmetries

of the product graph. By 1984 it was beginning to look like some much more general pleasantnesses concerning isometric embedding in products lay behind the previous results. Let us define a metric represen-

tation of a connected graph G to be an isometric embedding of G in a product H,O ::- OW,

201

Winkler: The metric structure of graphs

which is "irredundant" in the following sense: (1) each H; is a connected graph with at least two

vertices, and (2) every vertex of each factor appears as a coordinate in the image of at least one vertex of G. (It is not hard to show that any isometric embedding in a product can be made irredundant by discarding unused vertices and trivial factors.) Two (metric) representations are equivalent if there is a bijection between the factors of one and the factors of the other, together with isomorphisms between corresponding factors for which the obvious diagram commutes. A graph G is said to be irreducible if all of its representations are trivial, that is, equivalent to the identity map from G to itself. All of this is of course a parody, in a sense, of subdirect products in algebra, but as far as we know no algebraic interpretation is available. Nonetheless everything one could hope to be true actually is.

Theorem

(Graham & Winkler (1984,1985)). Every connected graph G has a unique canonical

representation tion

G — G,O :-- OG, in which every factor is irreducible. For any other representa-

G > H,O---OH,,

there is a surjection ¢: {1,...,4} — {1,...m]} between the index sets

and representations H; —-> []{G;: (j)=i} for which everything commutes; that is, the canonical

representation can be factored through any other. Furthermore, the canonical representation of G can be obtained by a polynomial-time algorithm.

Proof: Fix a representation

G + H,O ---OH,

of G and denote by (v;,...,V,_) the image of

a vertex v of G. Let e={x,y} be an edge of G with Nxy, xNy and Nyx (defined above) partitioning V(G). (See Fig. 2.) If e belongs to the ith factor, that is, if x;#y,;, then of course Xj=)j for all j# so for any vertex v,

dg (vx )}-de (vy) = dy. (vii -dy,(¥i,¥i) It follows that the location of v in the above partition depends only on v;. If f ={u,v} is an edge which crosses the partition, e.g. ueNxy and vexNy, then u;#v; and therefore f and e belong to the same factor.

Define the relation 6 on E(G) as follows: if e={x,y} and f ={u,v} are edges then

202

Winkler: The metric structure of graphs

Fig. 2

203

Winkler: The metric structure of graphs

eOf iff d(x,ujtd(y,v)

204

#d(x,v}+d(y,u) .

In that case f indeed crosses the partition induced by e, so e and f belong to the same factor. The relation @ is reflexive and symmetric but not generally transitive; let 6 be its transitive closure. Then e bf again implies that e and f belong to the same factor. If the equivalence classes of 6 in E(G) are E,,E,,...,E, then we already know that m1.

On the other hand, let G be the complete graph on vertices a,b,c,d with the edge {a,d}

Winkler: The metric structure of graphs

removed. If Player I is sitting on vertex b and sees c,a,d,a ,d,...,a,d in the window, he must go to c if the first unseen request is c but Stay at b if that request is a b. Hence this graph has infinite windex. It is not surprising, in view of previous comments, that the property of having windex