Combinatorial Set Theory

COMBINATORIAL SET THEORY

Neil H. WILLIAMS University of Queensland, Australia

1977

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK . OXFORD

@NORTH-HOLLAND PUBLISHING COMPANY - 1977 All rights reserved. No part of this publication may be reproduced. stored in a retrieval system. or transmitted. in any form or by any means. electronic, mechanical, photocopying. recording or otherwise. without the prior permission of the copvrighr owner.

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Library of Congress Cataloging in Publication Data Williams, Neil H. Combinatorial set theory. (Studies in logic and the foundations of mathematics ; v . 91) Bibliography: p. Includes indexes. I. Combinatorial set theory. I. Title. 11. Series. QA248. W56 5 1 1 ’. 3 77-3462 ISBN 0-7204-0722-2

PRINTED IN THE NETHERLANDS

PREFACE

Combinatorial theory is largely the study of properties that a set or family of sets may have by virtue of its cardinality, although this may be widened to consider related properties held by sets carrying a simple structure, such as ordered or well ordered sets. These properties are relevant to either finite or infinite sets, although frequently the questions that pose interesting problems for finite sets are either meaningless or trivial for infinite sets, and vice-versa. There has been a recent great upsurge in the study of finite combinatorial problems, and a significant, though more manageable, increase in interest in the combinatorial properties of infinite sets. This book deals solely with combinatorial questions pertinent to infinite sets. Some results have arisen purely in the context of infinite sets. One of the early results in the subject is the following: given an infinite set S of power K then there is a family of more than K subsets of S any two of which intersect in a set of size less than K . Chapter 1 looks at questions related to this. Problems of a different nature for families of sets are studied in Chapter 4, firstly a decomposition problem and then delta- and weak delta-systems. As a special case is the following: given any family A of Hz denumerable sets there is a subfamily 93 of A of size Hz such that the intersection of two sets from is the same for all pairs from CM. Chapter 3 is devoted to the study of set mappings, that is, functionsf defined on a set S such that given any x in S then f ( x ) is a subset of S for which x $ f ( x ) . Conditions are placed on t h e family { f ( x ) ;x ES } which ensure the existence of a large free set T ( a subset of S such that x 4 f ( y )and y 4 f ( x ) for all x,y from 2‘). Other results stem from problems that have been extensively studied for finite sets, and have been found to yield interesting questions when reformulated to apply to infinite sets. For example, in Chapter 5 we study infinite graphs, and in particular show that for any infinite cardinal K there is a graph with chromatic number K which contains no triangle, or indeed no pentagon; however any graph which contains no quadrilateral has chromatic number at most KO. Chapters 2 , 4 and 7 are devoted to various extensions of Ramsey’s classical theorem, which (in its finite form) states: given integers n, k, r if the

VI

Preface

/r-clcnient subsets o f a linitc set S are divided into r classes then provided that

S is sul’ficiently large there will be some k-eleinent subset of S all the n-element

sulwts ol’which l’all in the one class. Chapters 2 and 4 cover ordinary partition iclations, polarized partition relations and square bracket partition relations 101cardinal numbers, whilst Chapter 7 is concerned with ordinary partition relations f o r ordinal numbers. Familiarity with the standard notions of set theory has been assumed throughout. An Appendix summarizes those properties of cardinal and ordinal numbers and their arithmetic which are basic t o a study of this book. The development of infinitary combinatorial theory over the past twenty ycars or so has been greatly stimulated by associates of the Hungarian school, under the encouragement of Paul Erdos in particular. A glance at the list of references gives some indication of how many of the results in this book show his influence. T o all those people who have created this subject, I here acknowledge my debt and record my gratitude. Brisbane. 1976.

Neil H. Williams

FOREWORD ON NOTATION

The following is a brief summary of standard notation in use throughout this book. More technical notation is introduced throughout the text as the need for it becomes apparent. The Index of notation (pp. 206-208) provides a ready reference to the page on which a symbol is first defined. Set membership is denoted by €, and C is the inclusion relation with C denoting proper inclusion. The set of all subsets of a set is 'Ipx,so 8 x = { y ; y C_ x }. The union of all the sets in x is written Lbc and the intersection n x , so u x = {z;

3E

~ E (~ ) ~}nx ; = {z;

v ~ EE ~ ) .(} ~

Set difference is written x - y , sox - y = { z E x ; z e y } . The set of unordered pairs, one member from A and the other from B is A o B , so A o B = { {x, y }; x E A, y E B and x # y }. Ordered pairs are written ( x ,y ) , and sequences as ( x a ;a! < /3). The length of the sequence ( x a ; a! < /3) is /3, written Pn(x, ; a! < /3> = 8. If x , y are two sequences then x"y is the concatenation o f x a n d y , that is, the sequence obtained by placing the entries fromy in order after those from x. If A is any set, then the domain and the range of A are defined by dom(A)

= {x;

3 y ( ( x ,y ) € A ) ) ;

ran(A) =

b;3x((x,y ) € A ) } .

That f i s a function with domain A and range contained in Bis indicated briefly by writingf: A + B. The set of all such functions is *B, so

* B = {f;f:A+B}. Iff: A -+ B , then the value off at x is f (x). The restriction off to a set X is writtenf r X , so

frx={

( X , ~ ) E ~ ; ~ E X ) ;

and f [ X ]is the range off

r X , so

f[XI = { f ( x ) ; xE X ) . The words set and family are used synonymously. However an indexed family, written (Ai; i E I ) , stands for that function A with domain I and

Foreword on notation

X

A(i) = Ai for each i in I. The Cartesian product of an indexed family is written X(Ai; i E l). A decomposition of a set A is a family A of sets such that A = UA; this is the same as a partition of A. The partition A is disjoint if the sets in A are pairwise disjoint. For elements a, b of A and a partition A of A , the notation

a

b(mod A )

means that there is some Ak in A such that a, b E A k . The cardinality of a set X is written 1x1,and [XIK, [XIcK, ... denote {Y C X;IYI = K }, {Y C_ X ; IYI < K } , ... .The operations of cardinal addition, multiplication and exponentiation are written q i6',q . 6' and q', while the corresponding ordinal operations are written a + 0,@ and.!a The infinite cardinal sum and product of an indexed family (qi;i E 1) of cardinals are written C(Q~;i E Z) and n(qi;i E l ) , whereas the ordinal sum and product of a well ordered sequence (a,,; v < (3) of ordinal numbers are written &(a,,;v < 0) and llo(a,;v < /3)Let X be a set ordered by a relationh,.

Then if E.C # v, we have g,(a) #gv(a) whenever a > max(A,, A), and so g l , ng,l < h,,,axb,v)< A. Then 9 = {g,; K < v < K ' } has the properties claimed. To establish that III G K", suppose for a contradiction that we have a family 9 of almost disjoint functions in ' K with 191>K+++. The argument goes as in the proof of Lemma 1.2.3, taking the decomposition

only using now the relation K+++ +. (K+):+ which is a consequence of Theorem 2.2.4. This ensures that there is a family 9 with 9 E [TIK+ such that for

Ch. 1.2

Almost disjoint functions

9

some fixed 7 with T < A' we have ( f ngl < A, for a l l 5 g in 9. But now if 8( = {g r A:;g€ g}, one finds 1911,=K + and 6(8(< &, contradicting Lemma 1.2.2. This leaves (i), so suppose henceforth that A' # K ' , K + . If K + < A' then 111 = K from Lemma 1.2.2. If A < K , by considering any family 9 where 9C ' K as a decomposition of A X K into sets each of power K , when 6 ( 9 ) < A by Corollory 1.1.7 171< K and consequently III = K . Only the situation when K < A and A' < K + remains, and here A is singular. Choose regular cardinals A, for u with u < A' such that K++ < A-, < A1 < ... < A and h = Z(A,; u < A'). In fact A' < K , for A' = K would mean A' = A'' = K ' since always A' is regular. We consider separately the two possible cases, A' < K ' or A' > K ' . The constructions that we use come from the paper [70] of Milner. Take first the case A' < K ' . We have claimed in the statement of the theorem that nt = K . For a contradiction, suppose we have a family 7 consisting of painvise almost disjoint functions in ' K with 171= K + . When a < A and 0 < K , put ?ao = {f€ ? ; f ( a ) = 0). Then for each a with a < A, the sequence ( 9 B~< ; K ) gives a disjoint decomposition of 9 into K classes. However the number of such decompositions is K " ' (since each element of 9 may be in any class), thus the number of decompositions is K ~ =+ K + + . Since each A, is regular with K++ < A,, for each u there must be a set Mu in [A,] ' 0 such that the decompositions 0 < K ) are the same for all a in Mu.But then for alr az in Mu,for all f i n 7 we have f ( a l ) = f(a2).Thus each f i n 9 induces a map f* : A' + K , where

f*(u) = f ( a ) for any a in Mu . However, the number of maps from A' to K is K " , and this is just K since A' < K ' . We had assumed that 191= K ' , and so there must be distinct fl, fz in 9 with ft = Thus fl (a)= f2 (a)for all a in Mu,for any u with u < A'. This means

fz.

If1

nf2l2 lU{M,;u
K ' , is similar but rather longer. Again the claim is that 111 = K , and again we seek a contradiction from a family ?of K + almost disjoint functions in ' K . Since here K' < A' < K , also K is singular and we may, choose regular cardinals K , where 7 < K ' such that K~ < K < ...< K and K = Z(K,; T < K ' ) . When a < A and 7 < K ' , put ?(a,7)= {f€ 9; K, of cardinals, what is the size of the largest almost disjoint family of functions taken from the Cartesian product X(K,; u < A) of the K,? (Thus families from "K correspond to the special case where K, = K for all u.) This new situation is particularly relevant if K is a singular cardinal, A = K ' and the K, form a sequence of cardinals below K converging to K . There is the following lemma which will enable us (in certain circumstances) to construct large almost disjoint subfamilies of a Cartesian product.

'

Lemma 1.2.8. Let (A,,; v < 77) be any sequence of cardinals and let (A,,; v < 77) be any sequence of sets with always &I = It@,; P G v). lien there is a subfamily 9ofX(A,; v < 77) with 6(9) < 7) and 191= II@,,; v < 77).

Proof. For each v, identify A , with X@,; 1.1< v). Put 'II = II@,,; v < q), and let (fa;(Y < n) be an enumeration of X(A,,; v < 77). Define functionsg, in X(A,,;

Ch. 1.2

Almost disjoint functions

u < 0 ) for a with a < n as follows: if v

11

< 17 then

r (V + 1). a < n} so 171= n. Further 7 is almost disjoint, for take distinct

gOlw= f a

Put 9 = {g,; functionsg,, gp from 9. If u is the least ordinal such that f,(u) # fp(u) then g,(p)#g&)wheneverp>u.Thus Ig, ngpl< Iul x ( K n

; n < W)'

K

.

The result in Corollary 1.2.9 is special to singular cardinals K with K' = No. When K is singular with K ' 2 HI,for some increasing sequences ( K ~ u; < K ' ) of cardinals below K there is an almost disjoint subfamily 9 of X ( K ~ u; < K ' ) with 1 9 1> K ; for other sequences possibly 191< K for all such families 9. Examples are given in Theorems 1.2.15 and 1.2.13 below. However, first we need to examine some properties of closed unbounded subset;. of the cardinal h.

Definition 1.2.10. Let h be an infinite cardinal. The subset C of h is said to be closed if whenever B E [C]"' then U B E C. The subset C of h is said to be unbounded (or cofinal) in h if for all a with a < h there is y in C with y 2 a. If h is regular, it is easily seen that the intersection of any family of fewer than h closed unbounded subsets of h is again closed and unbounded in A.

Definition 1.2.11. A subset S of h is stationary in h if S intersects every closed unbounded subset of A. It is clear that any stationary subset of a regular cardinal h has power h. We shall make use of the following result of Fodor [42].

Lemma 1.2.12. Let S be a stationary subset of the uncountable regular cardinal h and let f : S -+ h be a regressive function (that is, f ( a )< a whenever a # 0). Then there is a stationary subset T of S such that f is constant on T.

Ch. 1.2

Almost disjoint families of sets

12

Proof. Suppose the lemma is false, so for all 0in S the set f-'(fl) is not sta. tionary in A. Hence there is a closed unbounded subset Cp of A such that cpnf-l(o) = 8 . Put D = {a A for more than A values of p with p < v, and so perhaps !A n U{T,; p < v}l > A, meaning that there would be no suitable extension of (U { T,; p < u } ) to all of d.To prevent this happening, we should finish with such an A at the first stage v where IA n (U{X,; p < v})l> A. So at any stage v, we treat also the members of 93, where 93, includes all members of d with IA n U{X,;p < v}l > A, and form Xu from d , U 73,. The formal construction proceeds as follows. Define sets Xu where v < L' with Xu E [ U d]" by induction as follows. Suppose for a particular v with v < 1' that the X, where p < v have already been defined. Put X,* = U{X,; p < v}, so IX,*l< 1. Write 9, = { A € d ;IA n X,*l> A}. Since 6(d)< A, any A in 73, is uniquely determined byA nX,*.Since IX,*l 1,2 0, nothing more need be done. If k = 0, then [H]" C u{ A l ; l < 6 ) and -+

26

Ch. 2.2

Ordinary partition relations

because of the relation qo + (81;I < 6)" there must be 1 with I < 6 and a subset I of H with 1 1 1= O r and [I]" C Al. In any event, there is a suitable subset of S homogeneous for the original partition of [S]". The restriction to finite exponent n is justified by the following theorem from [27], which shows that the weakest ossible non trivial partition relation with infinite exponent, K (No,No)L, is false for all K . -+

Theorem 2.1.3. Given any infinite set S, there is a partition [S]"O = A0 U A1 of the denumerable subsets of S into two classes which has no infinite homogeneous set. Proof. Let < be a relation which well orders [SIHo. Define a partition [Sf' = A, u A, as follows: for x in [slHo,

XEAO*~YE[X]~~(Y<X), X E A1 -vY E [XIHo( X S Y) . Take any set H in [SINo; we shall show H is not homogeneous for this artiR tion. Let X be the least infinite subset of H ; clearly X E A, and so [H] A,. On the other hand, write H = { h i ;i < a} and for each finite k , put Hk = {ho, hz, ..., h2k } U { h ~ i ;+i