Chapter 7. Ordinary Partition Relations for Ordinal Numbers

CHAPTER 7

ORDINARY PARTITION RELATIONS FOR ORDINAL NUMBERS $ 1. Introductory remarks As noted in Chapter 2 , the definition of the ordinary partition relation for cardinal numbers can be extended to order types. In this chapter we shall consider the partition symbol for well ordered types, that is, for ordinal numbers. The definition is as follows.

Definition 7.1.1. Let a,y,ak (where k < y) be ordinal numbers and let n be a positive integer. The ordinary partition symbol a + (ak;k < 7)" means the following. Let S be a set ordered with order type a. For all partitions A = ( A k ; k < y} of [S]" into y parts, there exist k with k < y and a subset H of S having order type ak such that [HI" C A k . The various conventions concerning the use of the partition symbol adopted for cardinal numbers in Chapter 2 will be followed without further comment for ordinal numbers. The problems concerning the symbol for ordinal numbers are considerably more ramified than those for cardinal numbers. We shall confine most of the discussion to the case n = 2 , and frequently y = 2 as well. Even so there are many unsolved problems, and we shall not attempt to cover even all the cases where progress has been made. We shall limit our treatment to a few special Cases where a reasonably complete discussion is possible. Thus 8 2 is devoted to partitions of [wN12into two classes, mainly for finite a. In $3, we prove Chang's theorem for w w .And in $4 we shall consider relations of the form w1 + (a1,...,ak)', and prove that w 1 + (a): for countable a and finite k. In this chapter we shall need to distinguish clearly between the order type of a well ordered set and its cardinality. The problem is particularly acute with the initial ordinals. When order type is to be emphasized, we shall write

154

Ch. 7.1

Ordinary partition relations

w , w l , w 2 , ..., a,, ... for the sequence of infinite initial ordinals, although the sequence Ho,Hi, HZ, ..- H,, ... of infinite cardinals denotes the same se-

quence (as a sequence of sets). It will be left to the context to distinguish between the symbols $ for ordinal exponentiation and K~ for cardinal exponentiation. In particular, s mbols such as a : , &, ... stand for the ordinal . . operation, whereas Hi,H, , ... indicate cardinal exponentiation. The symbols Co, II, are used to indicate the ordinal sum and the ordinal product of a well ordered sequence of ordinals. It is easy to see that a partition relation between cardinal numbers is equivalent to the same relation between the corresponding initial ordinals. Unlike the situation with cardinal numbers, for ordinal numbers results with n = 1 are not trivial. In [73], Milner and Rado consider this situation. They give an algorithm to determine in finitely many steps for any sequence ( a k ; k < y) of ordinals the least a such that a .+ (ak;k < y)'. We mention a couple of results from [73], but otherwise refer the reader to the original. There is the following lemma.

x,

Lemma 7.1.2. (i) Suppose a + ( a k ; k < y)' and /3 ( P k ; k < y)'. Then 09 ( a k P k ; k < 7)'. (ii) Let y be finite. Suppose a,, (a,,k; k < y)' for p with p < p. Put a@)= ~o(cu,;P < P ) and a&) = no(a,,k; p < P). Then a ( p ) (a&); k
-+

-+

+

Proof. A special case of (i) was stated as Lemma 5.1.6, and tiic proof of the general case hardly differs from the proof of Lemma 5.1.6. Let S be a set well ordered with order type 09, so we may suppose S = X a under the lexicographic ordering. Take any partition S = U{ Ak; k < y}. For x with x < p put Ak(x) = { y < a;(x, y ) E Ak}. For each x , we have a = U{ Ak(x); k < y}, and so there is k(x) such that tp(Ak(x)(x)) 2 Put r k= {x < 0;k(x) = k}, so p = U( r k ;k < y). Then there is ko such that tp(rk0) = pko. Put R = {(x, v)E S;X E r k , and y E Ako(x)}. Then R L Ak, and tp(R) 2 @k,pk,, so tp(Ako) 2 a k O P k O . This proves (i). To prove (ii), for each u with u < p , put a(v) = n,(a,,; p < v ) and a k ( v ) = IIo(a,k; p < u). Use transfinite induction on p . The case p = 1 is trivial. Suppose p is a successor ordinal, say p = u t 1. Then a(u) -+ ( ~ ( u ) ; k < y)' by the inductive hypothesis, and a, -+ (a,k; k < y)'. So by (i), a@) (a&); k < 7 ) ' .Now suppose that p is a limit ordinal. Let S be a set well ordered by a relation < with order type a@), and suppose S is partitioned, S = u{Ak; k < y}. For each p with p < p there is a subset T,, of S with tp(T,,) = a@).By the inductive hypothesis there are always k(p) and a subset H,, of T,, with tp(H,,) 2 akcr~)(p)and H,, C Aku). Put r k= { p < p ; k(p) = k}, +

Ch. 7.1

155

Introductory remarks

so p = u{ r k ; k < y}. Since y is finite and p is a limit ordinal there is some ko such that r k o is cofinal in p. Then for each p in r k o we have tP(Ak0) = tP(Ak@)) Hence tp(&,)

ak@)(p)= ako(p> .

tp(H&)

2 SUp(ako(/l); /l E r k o ) = olko@). This COmpkteS the proof.

Theorem 7.1.3. If m is finite then w"

+

(w")h.

Proof. From Lemma 7.1.2(ii), noting that w + (a);. The second, somewhat surprising, result from [73] concerns partitions into infinitely many classes. It has been referred to as the Milner-Rado paradox.

Theorem 7.1 A. For all 0, $a

< up+'

+ (up"; k < a)'

then a

Proof. (Note that trivially up+' + (wp+l):, so certainly up+'+ (up"; k < a)'.) It suffices to show that if a < up+' then w; (up"; k < w)' . This we prove by induction on a.The case a = 1 is trivial. Suppose a is a successor ordinal, a = y + 1, and uJ (up"; k < w)'. Take any set S ordered by a relation < with order type a;, so S = U{S,,; 1.1 < wp} where tp(S,) = wJ and S,, < S, whenever p < v < u p . By the inductive hypothesis, for each p there is a decomposition S,, = A,,k; k < w ) where tp(A,k) < up". h t A0 = A1 = f$ and for k with k 2 0,

+

+

u{

Ak+2 = Then S =

u

AMk ;P < u p 1

u { Ak; k < w } and tp(A0) = tp(A1) = 0 ,

tp(Ak+,)

< &(Opk; p < U p ) =

Ur' <

W r 2

.

+

This gives a partition of S which demonstrates w$ (up"; k < a)'. Now suppose a is a limit ordinal and that wJ (wpk;k < a)' whenever y < a. There is a sequence (a@);p < up>of ordinals below CY with a(0)< a(1) < a(2) ... such that a = sup {a@); p < u p } , Let S be a set ordered with order type ZO(w;@);p < up), so t p ( 9 2 w;, and write S = U{S,,;p < up}where tp(S,,) = w j @ )and S,, < S, whenever p < v < wp. By the inductive hypothesis, for each p there is a decomposition S,, = U{A,,k; k < w } where tp(A,,k) < wpk. As before, put A0 = A1 = 8 and for k with k 2 0,

+

Ak+2 = U{Apk;P
156

Ch. 7.1

Ordinary partition relations

We shall conclude this section by noting a couple of negative relations.

Theorem 7.1.5. For all P, $ a


then a

+

(up + 1

ul2

Proof.If Icy1 < Np this is clear, so suppose Jal= Np. Let < b e a well ordering of a of order type u p . Define a partition [aJ2= A, U A, by: if u, T < a with 0<7,

{u,T)EA,*u
{u,T}EApu>T.

Take a subset H of a.If [HI2 C A, then both < and Q agree on H, so tp(H, <) = tp(H, q;hence tp(H, <) < up. If [HIz C_ A, then H enumerated in increasing Corder gives a descending <-chain of ordinals; hence H is finite. Thus this partition of [.I2 suffices to prove the theorem. Results of Kruse [ 6 2 ] extend Theorem 7.1.5 as follows for values of n with n 2 3.

Theorem 7.1.6. Suppose n 2 3 . For all 0, ifa

< up+'

then a f . (a0t 1, n + 1)".

+

Proof. By Theorem 7.1.5, a ( u p + 1, a)"-'. Thus there is a disjoint partition A = {A,, A,} of [a]"-' such that there is no subset A of a of order type upt 1 with [A]"-' C A,, nor an infinite subset B with [B]"-' C A'. Define a disjoint partition r = { ro,PI) of [a]" as follows: if u1 < (12 < ... < u,, < then {al,..., u , ) E r o * {ul,..., u,-l)EA, or (02 ,..., o , } E A o , {u, ,..., u , ) E r l *

{u1,..., ~ , , - ~ ) E A , a n d {u2 ,..., u , ) E A 1 .

Take H from [a]"", say H = { ul,..., u,+'}, listed in increasing order. If both { u,, ..., un} E rl and { u2, ..., a,+,} E rl we would have the contradiction { u2, ..., un} E A, n A,, so [HI2C rl. Suppose there is a subset H of a with tp(H) = u p + 1 such that [HI2C ro. Put

B = {u E H ; for all {al, ..., u n n - l from } [HIn-', if 01,

..., un-.l G o then {ul, ..., un-,) € A l } ,

so [B]"-' C A,. Put A = H - B ; then B < A . And in fact [A]"-' C A,. For take T ~ ..., , T ~ from - ~ A where < 7 2 < ... < ~ " - 1 . There are u1, ..., unVl in B with u1 < ... < un-l G and { ul, ..., unPl} 4 A'. Let u1, ..., u, list { ul, ..., un-,} U { 71, ..., T"-,} in order (so m = 2n - 2 or 2n - 3 depending

157

Countable ordinals

Ch. 1.2

on whether u n - ] < r 1 or un-l = rl). Always { ~ f + ..., ~ ui+"} , E ro, and { ul, ..., un-l} 4 A1. Hence an easy induction on i shows that always (ui+2, ..., ~ i +E~A,. } In particular (71, ..., T n - i } E A,, so indeed [A]"-'C A0 as claimed. Now tp(B) + tp(A) = tp(H) = u p + 1, so either tp(B) 2 w or tp(A) 2 u p + 1. This contradicts the choice of the partition A. Hence if H C a with tp(H) = w p + 1 then [HI2 ro.Thus the partition r of [&In suffices to prove the theorem. $ 2 . Countable ordinals

In this section we shall discuss partition relations of the form a-+(a~,al)" where a is a denumerable ordinal. From Ramsey's theorem, a (w, w)". By Theorems 7.1.5 and 7.1.6 (with /3 = 0), a f . (w + 1, and a f . (w+l, n+l)" if n 2 3. Hence the only relations of interest are those of the form a+(ao,m)2 where m is finite. Some of the first such relations to be established are in Erdos and Rado [ 2 9 ] ,where it is shown that wm (w + I, m)2 and wm f . (o+ 1, m + Moreover, for each k and m it is shown that there is a least integer Zo(m,k ) such that wlo(m, k ) (wk, m)2,and that a f . (wk, m)2 if a < wlo(m, k). In [ 3 2 ] ,this result is generalized to arbitrary /3 by showing that there is a least integer Zp(m, k ) such that oplp(m, k ) -+ (wok, m)2, and that a f . (opk, m)2 if a < oplp(m, k). Erdos and Rado conjecture in [ 3 2 ]that in fact Ip(m, k ) = lo(m, k ) for all /3. This was later proved correct by Baumgartner [ 3 ] .The reader is referred to the papers mentioned for the proofs of these results. We shall concentrate on partition relations for off, the ordinal powers of w, with countable a.The first results will be concerned with w" for finite n. For each positive integer n, put -+

-+

-+

W(n)= { ( a o , ...,

" w ; a O< a l

< ...
,

and let < be the lexicographic ordering of W(n), so (ao, . . . , a n - l ) < (bo, ..., b,-l)*ao
or (a0 = bo and a1 < b l ) or

... .

Under this ordering, W(n) has order type a".For a sequence a from W(n), the i-th component of a will be denoted either by ai or a(i), and similarly with b, c, ... .

Definition 7.2.1. A subset S of W(n) is said to be free in the i-th component

Ordinary partition relations

158

a.1.2

if for all (ao, ...,an-,) in S there is an infinite subset A of o such that for all a in A there are ..., a;-1 in w such that (ao, ..., q - 1 , a, a:+l. ..., aL-1) E S. Lemma 7.2.2. Let S be a subset of W(n)and take m with m < n. Then tp(S) > urnifand only if there is a subset Tof S which is free in m different components.

Proof. Suppose there is T with T C S such that T is free in m components. We shall show tp(T) 2 urn(and so also tp(S) 2 urn) by induction on m. This is trivial form = 1. So take T, free in m + 1 components. Let i be least such that T i s free in the i-th component, and letA be the infinite set of i-th components given by the definition. For each a in W(i t 1) which is an initial segment of a sequence in T, the set T(U) = { b E T; b r (i t 1) = a } is free in m components and so by inductive assumption tp(T(a)) 2 w". Since tp(T) 2 &(tp(T(a));a(i) € A ) thus tp(T) 2 wrnw = urn+'. Now suppose tp(S) 2 urn, and use induction to show that there is a subset of S free in m components. We may suppose tp(S) = urn. If m = 1, let i be maximal such that there is some sequence a of length i for which the set ~ ( a ) ={ E S ; ~r i = a } is infinite (where i = 0 and a = 8 is allowed). For such an a, put A = { bi; b E T(a)}; then the choice of i ensures that A is infinite. Thus T(u) is a subset o f S free in the i-th component. Now suppose the statement is true f o r m , and take a subset S of W(n) with tp(S) = urn+'. Let i be the largest integer such that there is a sequence a in W(i) for which the set

T(u)= { b ~ ~r i ;= ab} has order type a"". (Again i = 0 is allowed). So by the choice of i, for such an a , if

T,(u) = { b E T(u);b j = X} then tp(T,(a))

< urnfor each x in a.Put

X = { x E o ; tp(Tx(a))=wm}, then X is infinite (since urn+' = tp(T(a)) = &(tp(T,(a));x E a)). By the inductive hypothesis, each T,(a) for x in X is free in m components. Hence there is an infinite subset A of X such that all the T,(u) for x in A are free in

Ch. 1.2

Countable ordinals

159

the same m components. But then if T = U{ T,(u);x € A } , it follows that T is free in the i-th component and m later components; thus T is a subset of S free in m + 1 components. This completes the proof. The first paper in which partition relations for anare proved is Specker [93], where it is shown that w2 (a2, m), for every finite m, and un (an,3 ) , if n 2 3. The following simple proof of the first of these relations is from Haddad and Sabbagh [50]. -+

Theorem 7.2.3. For all finite m, the relation w 2 + (u2, m ) , holds.

Proof. Since tp(W(2)) = w 2 , it suffices to consider partitions of [W(2)I2. SO let any partition [W(2)I2 = A, U A 1 be given. Define a partition of [wI4 into 16 classes, [wI4 = u{A(io, ..., i3);io, ..., i3 = 0 , l } , by: for a, b, c, d in w with a < b < c < d,

{a, b, c, d } E A(io, ..., i3) 0 { ( a ,b),(c, d)} E Aio and { ( a ,c), (b,d)} E Ail and

{(a, d), (b, c)} E Ai, and {(a, b),(a, c ) }E Ais . By Ramsey's theorem there is H in [w]"O such that H is homogeneous for C A(io, ..., i3). Let ( h k ; k < w ) enumerate H this partition of [wI4, say [HJ4 in increasing order. If ij = 1 for any j , the following gives a subset I of W(2) of power m with [I], C A , : { (h2k, h 2 k + l ); k < m} if io = 1 ,

{ ( h k ,hrn+,J;k < m} if il

= 1,

{ ( h k ,h 2 r n - k ) ; k < m} if i 2 = 1 , {(ho,hl+k);k
I = { ( h , h ' ) ; h E H oa n d h ' E H h + l a n d h < h ' } . Then I C W(2), tp(0 = w 2 and [ I ] , C A,. Thus the theorem is proved. Lemma 7.2.4. The relation w 3

-+

(a3, 3), is false.

160

Ch. 1.2

Ordinary partition relations

Proof. Consider the following disjoint partition [W(3)I3 = ro U rl where {a,b}EI'l * u ~ < u ; < b o < a z < b l


rl

rl,

Suppose [ { a , b , c}]' C where a b ic. Since { a , b } E certainly a2 < b l , f r o n i { b , c } E r l followsbl
<

>

>

> rl

[a2ro.

ro.

vides an example which proves the lemma. Theorem 7.2.5. If 3 < m < w then 0"

+ (am,3)2.

Proof. The case m = 3 is Lemma 7.2.4. Form with m > 3, it is enough t o prove the following: (1) there is a one-to-one m a p f : a"' -+ w 3 such that if X is a subset of urn with tp(X) = w"' then tp(f[X]) = w 3 . (In this situation, one says thatfpins wm to w3.) For suppose (1) established. Take a partition A = {A,, A,} of [a3]' such that there is no H in [w3I3 with [HI' C_ A,, nor H with tp(W = w 3 and [HI2 C A,. Define a partition A* = {A:, AT} of w"' by, for x, y in am,

f[a

Then H in [wmI3 with [HI2 C_ AT would give in [w3I3 with If[H]]2CAl, whereas if H C urnand tp(W = amwith [HI2 C A: thenf[H] C w 3 , tpCf[H]) = w 3 and [f[H]]' C_ Ao. Thus A* shows urn (urn, 3)2. We prove (1) by observing that for any integers k, I : (2) if wk can be pinned to w', then wk+l can be pinned t o w ' + l . Since any one-to-one map f : urn-'-+ w pins urn-*to w , two applications of (2) establish (l).To prove (2),suppose indeed that wk can be pinned t o w! Write wk+las a disjoint union, wk+l= U {Si;i< w } where tp(Si) = u kand Si
+

Ch. 1.2

Countable ordinals

161

Calvin and Larson [46] investigate just which countable ordinals can be pinned to w 3 (the use of “pin” in this context is due to them), and as a consequence they show: if a is a countable ordinal and a + (a,3)2 then a = 0, 1, u2 or a = ww’ for some 0. It was noted by Erdos and Hajnal [24, see Problem 61 that for each n and k (k 2 3) there is a least integerf(k, n ) such that unP (ak, f ( k , n))2.(So from Lemma 7.2.4,f(3,3) = 3.) The exact value off@, n ) is not known in general. However, Nosal [75] has s h o w f ( 3 , n ) = 2n-2 + 1. This result appears as Theorem 7.2.9 below. The proof depends on the existence of cartain canonical partitions of W(n).

Definition 7.2.6. Two pairs { a , b } , {a’, b ‘ } from [W(n)I2are said to be similar if for all i, j with i, j < n,

ai < bj * a: < b; and ai > bj * a;.> b; . A partition A of [W(n)]’ is said t o be canonical if for all { a , b } , {a’, b‘} from [W(n)12whenever {a, b } is similar t o {a’, b ’ } then {a, 6 ) 5 {a’, b’} (mod A). The existence of canonical partitions was first proved by Hajnal, and independently later by Calvin.

Theorem 7.2.7. Given any disjoint partition A of [ W(n)12 into finitely many classes there is an infinite subset H of w srtch that the induced partition on [Pn w(n)12 is canonical. Proof. L e t S be the set of all strictly increasing functions f : n -+ 2n. Forfin 7 and x from [uI2“, say x = {XO, XI, ..., ~ 2 1 }< ~ put - xf = (Xf(0) ..., X f ( n - l ) ) . Suppose A = {A,,, ..., A,_ l } . For each x in define a function Fx : [SI2+rn by

Fx({.6 g}) = i

{xfixg} E Ai

.

Since there are only finitely many functions mapping from [TI2t o m , by Ramsey’s theorem there is an infinite subset H of w such that all the sets x from have the same F,, say F, = F. Take any similar pairs { a , b } , {a’,b’} from [H“fW(n)I2. l There are x, y in [HI2” andf, g in -7 such that a = xf, b = xg,a’ = y f and b‘ = y r Since Fx = F,, thus { a , b } = {xf, xg} E A F ( { f , g }and ) {a’, b‘} = {yf.Yg} E AF({f,g}), so { a , b } { a ’ , b ‘ }(mod A ) as required. Let n be fixed, with n 2 3. We define the following subsets of W(n):for i,

162

Ch. 7.2

Ordinaiy partition relations

j , k with 0 Q i< j < k

< n, put

... and aiPl = bi-' and ai< ... < a k _ l < b i < ... < b j - l < a k < ... < a , _ l < b j <

ECk = { { a , b } E W(n);a.

= bo

and

Consider the disjoint partitionr" = {I':,

ry = u{E$k;0 < i < j <

... < b , _ l } .

G}of W(n), where

k < n}

This partition has several useful properties, which we collect in the following lemma.

Lemma 7.2.8. For this partition r"of W(n), (i) there is no subset H o f W(n)of order type u 3such that [HI2L I$, (ii) there is no subset H o f W(n)with IHI = 2"-2 t 1 such that [HI2C ry, (iii) there is a subset H o f W(n) with IHI = 2n-2 such that C G.

[a2

Proof. For (i), take any subset H of W(n) with t p Q = u3. By Lemma 7.2.2 we may suppose that H is free in three components, say i, j , k where 0 < i < j < k. Thus given a in H , there is b in H such that b T i = a t i and b(i) >a(k- 1). There is c in H such that c r k = a r k a n d c(k) > b( j - 1). There is d in H such that d r j = b r j and d( j ) > c(n - 1). Then { c , d } E EZk, and so [HI2 n ry # 0.This proves (i). To prove (ii), use induction on n. When n = 3, the partition r3is the Specker partition used to prove Lemma 7.2.4, so (ii) is true when n = 3. Suppose (ii) is true for a particular value of n. Take a set A from [W(n + l)]q+' such that [AI2 C rl" ; we wish to show that q < 2,-'. Let A = {ao, ..., listed in lexicographic order. Note that for a, b with a < b and {a, b } E rl , always a(z) < b(i). Further the definition of any EZ:' ensures that b(l)fa(n), for if j = 1 then a(n) < b( j ) = b( 1) and if j > 1 then b( 1) Q b( j - 1) < a(k) Qa(n). Hence there is I, with 1 Q 4 , such that

'

ao( 1) Q a'( 1) Q

... Q a[( 1)
1) Q ... < as( 1)

.

(1)

We show first that Z < 2"-2. Observe for a , b with a b and {a, b } E I'"' that if a(0) < b(0) < b( 1) < a(n) then also { a , b'} E I'y'l where b' = (a(O), 6(1), b(2), ...,b(n)). For if a(0) = b(0) this is trivial, and if a(0) < b(0) then {a, b } EE:$ for some j , k , where j > 1. But then { a , b'} EEfh', so indeed {a, b'} E q + l . n u s if

A' = { (ao(O), am(l), ... a,(n),; )

then [A'I2 C I'"'.

Consequently, if

A" = { ( a m ( l ) ,..., a,(n)>;m Q Z }

112

< I}

Ch. 7.2

163

Countable ordinals

then [A"I2 C I'l. So by the inductive hypothesis, I t 1 < 2"-'. Return now to (1). Thus if 1 = 4 , nothing more is necessary. So suppose 1< q. Observe first that a1+1(0) = a,2(0) =

...= a,(O) .

(2)

For suppose (2) is false. Then there is rn with I + 1 < rn <(I such that
A* = {(a,&), ..., a,(n)>; I t 1 < rn < 4 ) ,

E.

as above [A*I2 C_ So again by the inductive hypothesis M*I < 2"-' , that is, 4 - I < 2"-2. Since I < 2n-2, thus 4 < 2"-'. This completes the inductive step, and proves (ii). Finally, we establish (iii). Again use induction; the case n = 3 is trivial. So suppose for some n that (iii) holds. The partition of { a E W(n t 1); a(0) = 0) induced by rn+'is isomorphic to the partition I-" of W(n), so.by hypothesis there is a subset H of W(n t 1) with IHI = 2"-' and a(0) = 0 for ail a in H such that [HI2 C I'y+'. Let H = {ao, ...,u p } , listed in lexicographic order (so p = 2"-2 - 1). Then


h t B = { b l ; O < l < p } a n d C = { c l ; O < l < p } . It iseasy tosee that {al,a,} is similar to both { b l , b,} and {cl, c,}. The definition ofr"" ensures that rn+lis a canonical partition of [ ~ ( +n1)12, so since c ry'l also [B]' C I'y" and [CI2 C I'"'. Moreover, for any I, rn with 1, rn < p there is k such that

[a2

bl(0) < ...< bl(k - 1) < ao(n) = ~ m ( 0 < ) bl(k) < ... < bl(n)


... < c m ( n ) , (1)

Ch. 7.2

Ordinary partition relations

164

and hence { bl, c, 1 E E:;;, so { bl, c,} € r'"'. Hence [B U C]' C r':", and since IB U C I = 2 n - 2 -+ ?"T2 = 2"-', the induction step is complete. This proves (iii), and comgletes the proof of the lemma.

Theorem 7.2.9. Zfn < 3 then w n f. (a3, 2"-2

+

and w n -+ ( a 32"-2)2. ,

Proof. From (i) and (ii) of Lemma 7.2.8, the negative relation is established by the partition r" of W(n). For the positive relation, take any disjoint partition A = {Ao, A,} of [W(n)I2such that there is no subset H of W(n) of order type w 3 with [HI2CAo and no 2"-2-element subset H of W(n) with [HI2C A , , and seek a contradiction. By Theorem 7.2.7 there is an infinite subset of w on which the induced partition is canonical, and we may suppose that A itself is canonical. For i, j , k with 0 < i
f.. (ao, a,, a 2 ) = (0, 11 k

1,

..., i-

1, i -+ ao, ..., j- 1 + ao, j + a,,

k - 1 + a l , k + a 2,..., n - 1

...,

+a2),

m f i j k ( a 0 , a l , a2) < . f i j k ( b o , b l , b 2 )9 (ao, a l , a 2 ) < (bo,b l , b2>.Define a disjoint partition A * = (A;, AT} of [W(3)I2by ( a l p 02, a 3 ) E

A;

* f i j k ( a l , a29 a 3 ) E A0

.

Clearly A * is canonical since A is. A subset of W(3) homogeneous for A* gives a subset of W(n)homogeneous for A in the obvious way. Hence A * is a canonical partition of [W(3)I2having no H of order type w 3with [HI2 C A,* and no H of size 2"-2 with [HI2 C_ AT. A check of the possible canonical partitions of [W(3)I2(see Milner [72]) shows that apart from the Specker partition r3 all have either H of order type w 3 with [HI2 in class 0 or else H of arbitrary large finite size with [HI2 in class 1 . Hence A* must be the Specker partition r3.Thus if (ao, a l , a 2 ) < (bo, b l , b 2 ) then {fijk(ao,al,a2),fijk(bo,

b1,bz)}€A,~ao
An easy check shows ( f i j k ( 0 , 1, n + l ) , f i j k ( n , 2 4 2n + 1 ) ) E E t k , and hence f 8. Since A is canonical, it follows that f$)k C A,. Since i, j , k were arbitrary, thus y'I C_ A,. Hence by Lemma 7.2.8(iii) there is a 2"-*-element subset H of W(n) with [HI2 C_ A , , contrary to the choice of A . This is the required contradiction, and the proof is complete.

E t k flA,

In particular, if we put n = 4 in Theorem 7.2.9, then we find u4-+ (0~,4)~ and u4 f . (a3, 5)2. These particular results were also obtained by Calvin, by

Countable ordinals

Ch.7.2

165

Hajnal (see [24]) and by Haddad and Sabbagh [Sl]. Earlier results for w4 were obtained by Milner [71], where it was shown that w4 + (a3, 3)2, w4 (w3 + 1, 3)2 and a! (a3, 3)2 if a! < w4. We conclude this section by proving the following theorem of Erdos and Milner [26]: wl+pm+ 2 9 ' where rn < w and p < wl. This result dates back to 1959; a proof also occurs in Milner [69]. The theorem does not give best possible results -- for example with p = 2, it gives w 2 m + i + ( w 3 , 2 ~ 2 whereas by Theorem 7.2.9 in fact + (a3, 2m)2 - but it seems to be the best general result of this type known.

+

+

Theorem 7.2.10. Let a!, /3 be countable, let k be finite. If a"+ (w"O, k)2 then we'@-+ 2k)2. Corollary7.2.11.1frn
+

k)2. Take any set S ordered Proof of Theorem 7.2.10. Suppose w" + and let a partition [SI2 = A0 U A1 be by a relation < with order type given. Assume there is no H in [S]wl+pwith [HI2 C A. and no H in [S]2k with [Ill2 C Al, and seek a contradiction. For any x in S, put Aj(x) = {Y ES; {x, U) E Aj}

*

We shall use the following observation. Suppose given a family { A , ; u < 6) where each A , is a subset of S of order type wQ.For x in S, put M(x) = { u < 6 ; tp(Ao(x) nA , ) = ma} . Then AE[S]We~tp{x€A;tp(M(x))=6}=w". Consider first the case when 6 is an ordinal power of w , say 6 = w y . Suppose (1) is false for a particular set A from [S]W~. Put A' = { x E A ; tp(M(x)) < w 7 } , so tp(A') = w". By the relation w" + (wl'p, k)2 and the choice of the partition, there is H in [A']ksuch that [q2 C_ A1. Since w7 + (a'>: (by Theorem 7.1.3) and tp(M(x)) < orif i E H, there is u with v < orsuch that u 4 U{M(x);x EH } . Thus for all x in H , tp(Ao(x) nA,)
Ch. 1.2

Ordinary partition relations

166

and [H U r]* C Al, contrary to the choice of the partition. Thus (1) holds when 6 = w7. For arbitrary 6 , write 6 as a finite sum of powers of w and successively use the above result finitely often to prove (1). From (1) we shall prove the following. Suppose given a family { A , ; v < u p } where each A , is a subset o f S of order type w" and moreover A,, < A , whenever fi < u < up.Suppose also finitely many values vo, ..., u, less than upare given. Then for any A in [S]"" there is x in A and an order preserving function g : up+. upsuch that g(uo) = vo, ...,g(u,) = v, and tp(Ao(x) fl Ag(")= w a whenever v < wp. (2)

For we may suppose vo < ul < ... < v,. Let A in [SIw" be given. Define subsetsBo, ..., B,+l o f S a s follows: Bo = A , B m + 1 = { X E B,;

tp(Ao(x)

fl A,,)

= a"}, (m = 0, 1,

...,n ) .

Successive applications of (1) with 6 = 1 show that always tp(B,+l) = wQ.In particular, if x E B,+l then tp(Ao(x) n A,,) = w" for each m. Define subsets CO,..., Cn+l of S as follows. Put Mo(x) = { v ; v < vo and tp(Ao(x) nA , ) = a"},

co =

{X E B n + 1 ; tp(Mo(x))

=~

09 )

and for m with m = 0, 1, ..., n put

M,+l(x) = { v ; u, cm+l

< u < v,+,

and tp(Ao(x) f A,) l = ma) ,

= C x E C m ; v m + 1 +tp(Mm+l(x))=vm+l)

9

(with the convention u,+~ = up). Successive applications of (1) show that always tp(C,) = war.In particular, tp(C,+,) = a" so Cn+lf 8. Choose x from for each m. Cn+l.Then tp(Mo(x)) = uo and v, + 1 + tp(M,+l(x)) = Hence if

M(x).= { v < up;tp(Ao(x) nA,) = w"} , then M(x) is the ordered union M(x)=Mo(x)u { v O } ~ M l ( x ) u{vllu ...U { ~ n } U M n + l ( x ) 9

and so tp(M(x)) = up.Let g : up+M(x) be the enumeration of M(x) in order. Theng(uo) = uB ...,g(un) = v,, and ( 2 ) is proved. Since lwpl = No, there is a sequence (7,;n < w>of ordinals below upin which every v with u < upis repeated infinitely often. We shall define by induction on the integer n elements x, of S and subsets A(n, v ) of S for v with

Ch. 7.2

Countable ordinals

167

v < u p . Write

s = U{A(O, v); v < UP} where always tp(A(0, v)) = oOrand A(0, p )
A*(% v) = { x E A ( n , v ) ; x > x , for all m with m < n } , then tp(A*(n, v)) = a".Use ( 2 ) on the family {A(n, v); v < up}with vo = yo, ...,v, = y, and A = A*(n, 7,) to find x , in A*(n, y), and an order preserving map g, : up-+ upwith g,(y,) = y m if m < n such that tp(Ao(x,) n A(n, g,(v))) = ooL whenever v < up.Put A(n + 1, v) = Ao(x,) nA(n, g,(v)). Then tp(A(n + 1, v)) = oOrand since g, is order preserving also A(n + 1, p ) < A(n + 1, v) whenever p < v < up.This completes the construction. Put H = { x n ;n < a}.The construction ensures that if n > m then A(n, v) C Ao(x,) for all v ; in particular since x , E A(n, yn) thus x , E Ao(x,). Hence [HI2 C A,. We shall show that tp(H) = u"~,contrary-to the choice of the original partition: This will provide the contradiction required to prove the theorem. Take integers m, n ; say m < n. Since

A(n, v) C A ( n - I,gn-l(v)) C A ( n - 2,g,-2g,-l(v))

c A ( m , gm ... gn-zgn-l(v)) in particular

9

c ...

168

Ch. 7.3

Ordinary partition relations

Now given m, the set { n < w ; m < n and y, = y m } is infinite. Further the definition of x, ensures that if p < n and yn = y p then x, > x p . Thus tp(H n A ( m , ' y m ) ) = w . Since { y m ;m < w } = UP, it now follows from (5) as claimed. This completes the proof. that tp(H) = wwp=

53. Chang's Theorem for ow One of the long-standing problems in the partition calculus for countable ordinals has been to decide whether or not the relation ww (ww, 3)2 holds (see Specker [93]). A positive answer was finally given by Chang [7] in a lengthy tour de force. His result was subsequently extended by Milner who showed that ww +. (aw, rn)' for all finite m. In [ 6 5 ] ,Larson gave a new short proof of this. We shall follow Larson's method. We shall make use of the following lemma, a consequence of Ramsey's theorem. -f

Lemma 7.3.1. Let S E [wIK0.Suppose for each a from there are given an integer m(a)and a partition A (a) = { Ao(a), Al(a)}l of [SIm('). Then there is an infinite subset H of S such thatfor each a from the set { h E H ; h > max(a)} is homogeneous for A (a). Proof. Inductively define setsHk from [sINo and elements hk offfk, for finite k , as follows. Put Ho = S, and let ho be the least element of Ho. If Hk and ho, ..., hk have already been defined, list all subsets a of {ho, ..., hk} with max(a) = hk, say a0, a l , ..., a[. Put Hko = H k and use Ramsey's theorem to choose successively infinite subsets H k i + l from H k i so that Hki+l is homogeneous forA(ai). h t H k + l = { h E H k [ + l ;h > h k } ; then H k + l is homogeneous for each A (ai). Let hk+l be the least element ofHk+l. Put H = {hk; k < w } . Take a from and suppose max(a) = hk. Then { h EH; h > h k } is a subset ofHk+l, and hence is homogeneous forA(a), as required.

Theorem 7.3.2. Let m be finite. Then ww +. (ow, m)2. As in the previous section, for positive integer n put

< ...< a n p l } , and order W(n) lexicographically. Put W = U{ W(n);0 < n < w } and order W ( n ) = {(a0,..., a , - l ) ~ " w ; a o < a l

the elements of W first by length of sequence, and then lexicographically in

Ch. 7 . 3

169

Change theorem for ww

each W(nj. Under this ordering, W has order type ww. So to prove Theorem 7.3.2, it suffices to take any partition [WI2 = A, U A , and show that, given m, either there is a subset H of W with tp(H) = ww such that [HI2 C A0 or else there is a subset H of W with IHI = m such that [HI2C A , . Let m be fixed. Take any partition A = { A o ,A , } of [WI2 for which there i s n o s e t H i n [W]" with [HI2 C A I . We must f i n d H i n [WIw" with

[HI2 L A,. By Corollary 7.2.1 1, certainly the relation w"" -+ (w", 2m)2 holds. By the relation applied to the partition of W(mn) induced byA , there must be a subset W'(n) of W(mn) with tp(W'(n)) = w" such that [W'(n)12C A,. Put W' = u{W'(n); 0 C n < a}, so tp(W')= ww and for { a , b } from [W'I2 with ln(a) = In@) we know { a , b } E A,. It will suffice to find a subset H of W' with tpQ = owsuch that for { a , b } from [HI2with In@) # In@) still { a , b } E Ao. By redefining the partition A , we may suppose that W' = W. For the rest of this section we shall use the convention that if a is a finite increasing sequence of integers then Q is the set of entries ofa and conversely, if a is a finite set of integers then a is the sequence of the elements of Q arranged in increasing order. The concatenation of sequences a a n d b is a b. c\

Definition 7.3.3. For a, 6 from Wwith In (a) < ln(b)and for integer k, say { a , b } has form 2k [or form 2k + I ] if there are non-empty finite sets of integers ao, al, ..., Q k and bo, bl, ..., bk-1 [bk] such that (i)

a O < b O < a l < b l < ...
(ii) a =ao "a1 "al

... ak ; A

b = b o " b l h...hbk-l for form 2 k ; b = b o A b l ^...^bk for form 2k + 1; (iii) if c = (laol, lao U all, ..., lao U ... U akl) and

d = (Ibol,Ibo U b,l,

..., Ibo U ... U bk-l

[U bk]l)then

c
Lemma 7.3.4. Let S E [wIK0.There is H in [SIN0,H = { h l ,h2, h3, ...I listed in increasing order, such that for all positive integers I there is i(1) = 0,l so that if { a , b } is any pair of sequences of form I and shape { c , d } with a, 6, c, d C { h E H ; h > h l } then ( a , b } E A,(,,.

Ch. 1.3

Ordinary partition relations

170

Proof. For each possible shape { c , d } with c, d C S and each a. in [S]<'O with c < a. < d, choose a partition A (c, ao, d ) of suitable size subsets of S into two classes so that: whenever x = a. U bo U a l U b l U ... where if a = a o " a l " ...,b = b o " b l "... then {a,b}hasshape {c,d}and { a , b } E A i , then x E Ai(c, ao, d ) . By Lemma 7.3.1. there is HO in [SIKosuch that for c, ao, d from Ho there is i(c, ao, d ) so that if a U b C { h E H o ; h > max(d)} and {ao "a, b } has shape { c, d } then (a0

"a, b ) E Ai(c,,o,d,

.

For each finite subset c ofHo, choose two partitionsA (c), A*(c) into two classes of the c(0) + IcI, c(0) + IcI - 1 size subsets of Ho so that: if x = a. U d where la01 = c(O), Id1 = IcI, c < a.

< d and

i(c, 110, d ) = i, then x E Ai(c) ; if x =ao U d where

11101 = c(O),

i(c, ao, d ) = i, then x

Id1 = IcI - 1, c
E AT(c)

< d and

.

Ap ly Lemma 7.3.1 twice to find H1 in [HolNO such that for c from [H,IR0 there are i(c),i*(c) such that for appropriate sized subsets a@d of { h E H1;h > max(c)}, if Id1 = IcI then i(c, ao, d ) = i(c) ; if Id1 = IcI - 1 then i(c, ao, d ) = i*(c) . Finally, define partitions A (2k), A(2k t 1) of [Hllk+' by c E Ai(2k) * i*(c) = i; c E Ai(2k

+ 1) * i(c)= i .

Use Ramsey's theorem repeatedly to find infinite subsets Hl+1 of HI minus its least element such that HI+^ is homogeneous for A (1) (where I2 1). Let hl be the least element of and put H = { h,; 0 < I < o}. Then for all positive 1 there is i(1) such that if c E [ ( h E H ; h > hl}lk+' then i*(c)= i(l)or i(c) = i(l),depending on whether 1 equals 2k or 2k t 1. It follows that H has the property required. For take a pair { a , b } of form I and shape { c , d } where a, b, c, d C { h E H ; h > hl). Since H C H1 C Ho, then { a , b } E Ai(c,ao,d) (where a. is the sequen,:e of the first c(0) elements of a), and

i(c, ao, d ) = i*(c) or i(c)(depending on whether I is even or odd) = i(l)

.

This completes the proof.

Ch. 1.3

Change theorem for ww

171

Lemma 7.3.5. Given any infinite subset H of w, for all positive integers 1, m there is M in [ W]" such that whenever a, b E M then a, b C H, { a , b} has form I and if { c , d } is the shape of {a, b} then c, d C H.

Proof. Suppose 1 = 2k or 1 = 2k t 1. Choose increasing finite sequences ci, U i j from H for i, j with 1 < i < m, 0
3

a12, a22, ..., U m 2 , ..., alk, ...,amk, arnk+l,.-)a2k+19alk+l

9

such that ln(ci) = k + 1, In(aio) = ci(0) and ln(Uij) = C i ( j ) - ci(j - 1) (for i, j with 1 < i < m , 1
A

M = { a i ;1 < i < m } . If 1 = 2k + 1 then { a i , aj} has form 1 and shape { ci, cj}, whereas if 1 = 2k then { a i , aj} has form 1 and shape { c i , dj} where dj = (cj(O), ..., cj(k - 2), cj(k t 1)). Hence M has the property required in the lemma. Lemma 7.3.6. Given any infinite subset H of w, say H = { h l ,h2, h3, ...} listed in increasing order, there is a subset X of W with tp(X) = owsuch that whenever a, b € X with In (a)
Proof. Choose increasing finite sequences ci, ai, a(i, j, k ) from H for i, j , k with 1 < i < w and 1
1, k ) , 4 2 , 1, k ) , a(2,2, k ) , 4 3 , 1, k ) , ..., a(k, k, k )

(that is, the a(i, j , k ) are chosen according to the lexicographic ordering of (i,j ) , where 1
112

Ordinary partition relations

Ch. 1.3

Put Xi =

{ai"a(i, 1, kl)=a(i, 2 , k2)" ..."a(i, i, ki); 1 < i
< k2 < ... < ki < o},

X = U{Xi; 1 < i < o } , S O XC W. For fixed i, all the sequences in Xi have the same length, namely

ci(0)t (ci(1) - ci(0)) t ... + (ci(i)- ci(i - 1)) = ci(i) . Further, the choice of a(i, j , k ) ensures that if k l

< k2 < ... < ki then

a(i, 1, k , ) < a(i, 2 , k2) < ... < a(i, i, ki) ,

and hence the order type of X i , in the lexicographic ordering on W,is wi. Moreover, if i h2*1; this ensures that c(0) > hl. Thus X has the properties required, and the lemma is proved. We are now set to prove Theorem 7.3.2, namely that ow+ (ow, m)' for all finite m.

Proof of Theorem 7.3.2. Take a partition [ WI2 = A, U A, for which there is no set H in [W]" such that [HI2 C A,. As noted before, we may suppose that whenever a, b E W with ln(a) = In@) then { a , b } E Ao. Take an infinite subset Hof w with the property given by Lemma 7.3.4. Thus i f H = {Al, h2, h3, ...}, listed in increasing order, then for every positive I there is i(l) = 0, 1 so that for all pairs {a, b } of form I , if { a , b } has shape {c, d } and a, b, c, d C { h E H ; h > hl} then { a , b } E A,(,). Given I, apply Lemma 7.3.5 to { h E H ; h > hl} to find M in [W]" such that all { a , b } from [MI2 have form I, and if the shape is { c , d } then a, b, c, d C { h E H ; h > hl}. Thus i(1)= 0, and this is true for all 1. Now use Lemma 7.3.6 to find a subset X of W with tp(X) = ww such that all { a , b } from X with In@) # In@) have form I for some I , and moreover if {c, d } is the shape 0 1 {a, b } then a, b, c, d C_ { h E H ; h > h,}. Thus { a , b ) E Ai(,), that is {a, 6) E A,, for all such {a, b } . Hence [XI2 C Ao, and the proof is complete.

Ch. 7.4

173

Partitions of / w 1 / 2

94. Partitions of [ a l l 2 A complete discussion is possible for the relation w1 + ( a k ; k < T ) ~ Some . results are immediate from Chapter 2. From Corollary 2.5.2 we know w1 (3):,, so only partitions of [ 0 1 1 2 into finitely many classes are relevant. From Theorem 2.5.8 comes the further negative relation w 1 ( ~ 1 ) ; .

+

+

Theorem 7.4.1. For sfinite,

w1

-+ (w1,

(w + I),)

2

.

Proof. Applying Theorem 2.2.6 to the two relations H1+ (Hl); and Ho+(Ho), 1 gives the relation N1 + (N1, Only slight changes to the proof of Theorem 2.2.6 yields the marginally stronger result w 1 + (wl, (w + 1),)2. Referring to that proof, with the notation in use there, the changes are as follows. Start by taking a well ordering < of the set S of order type K . When one comes to define the set F(v) for v in N n SEQ,,, put C(v)= { x ~ S ( v ) ; x < fyo r s o m e y i n U ( F ( V ~ T ) ; T < U ) } ,

and take for F(v) the union of G(v) and the original F(v). Since K is regular, IC(v)l < K and so still IF(v)l < K . The elements xu are defined as before, using a sequence v in N n SEQk2 for which S(v) # 0.Choose x in S(v). Then for each u it follws that x 4 F ( v r u + l), so in particularx 4 G(vru t 1). Sincex ES(v To t 1) and xu E F( v r u) it follows that x # xu;hence xu < x. From x E S(v u + 1) we know {xu, x} E r r ( v r o+l).It follows that HA = HO U {x} is a subset of K of order type at least qfo+ 1 with [HA]2 rro.

r

Theorem 7.4.1 is the strongest positive result of its type. The relation (w1, w t 2)2 was proved by Hajnal [52] assuming the continuum hypothesis. Whether the relation can be proved without this assumption is an open problem. See Erdos and Hajnal [24], Problem 8, and their discussion of this problem in [ 2 5 ] . w1

+

Theorem 7.4.2(CH).

w1

+

(017

w + 2)2.

Proof. Take any set S ordered with order type w1 by a relation<. By Corollary 3.2.8 there is a set mappingf: S-+CpS of order tE1 with If(x)nflv)l< Ho for any pair x, y from S, such that f h a s no free set of power H1.Since always If(x)l < Ho, we can choose inductively elements x, of S for a with a < w1 so that

a
andf(x,)<

{xp}.

174

Ordinary partition relations

Ch. 7.4

Put T = {x,; a < wl) so tp(T) = wl. Define a partition [TI2= A, U A1 by, if a < 0 then {x, xp) E A1

* x,

Ef(xp)

.

Any subset HO of T with [&I2 C A, is free forf, so tp(H0) < 01. Suppose there isH1 in [T]w+2with lH112 C A l . I f x a n d y are the o - t h and (a+ 1)-st elements o f H l thenH1 - {x, y ) C f ( x ) n f ( y )so If(x) n f ( y ) l = No,contrary to the choice off. So there is no such set H1,and the theorem is proved. We have left to consider relations of the form 01 -+ k < s ) ~where s is finite and the (Yk are all countable. Trivially Ramsey's theorem gives w1 (a),'. Erdos and Rado [29] showed that w 1 + (w + n); for any finite n. In [60], Hajnal strengthened this to w1 (w2, for any finite n. The next result was due to Calvin (1970, unpublished) that w 1 -+ (03);. Later still Prikry [79] obtained w1 + (a2+ 1, a)2 for every countable a. Finally Baumgartner and Hajnal [4] settled the problem by proving the best possible result, namely w 1 +(a),' for every cduntable a and finite s. In fact they proved somewhat more. They showed that if cp is an order type such that cp + (w)k then cp + (a),"(for all countable a and finite s). Their proof uses deep methods from mathematical logic. They show first that the relation cp (a),"holds in a particular model of set theory, and then use absoluteness criteria to conclude that the relation is true in the real world. Calvin [44] has since given a combinatorial proof of the Baumgartner-Hajnal theorem. We shall devote the remainder of this section to Calvin's proof for the relation -+

-+

-+

a 1

+

(4,".

Theorem 7.4.3. Let a be a countable ordinal and let s be finite. Then w 1 +(a)%. For the rest of this section, we adopt the convention that the letters a,0, y, ... (with maybe subscripts, superscripts or the like) range over only the ordinals less than w 1. Let the partition A of [all2 be given, [ a l l 2 = A0 U ... U As-l . For any a,we shall show that there is H in [ollw" such that H i s homogeneous for A ; from this Theorem 7.4.3 follows. We introduce the following property, somewhat weaker than being a homogeneous set for A.

Ch. 1.4

115

Partitions of / w 112

Definition 7.4.4. A subset X o f w l is almost homogeneous (for A ) if wheneverA € [XIw" (for any a) then for all p with < a there are B in [A]"', A' in [Alwa and i with i < s such that B < A ' and B e A' C_ Ai, where B e A' = {{x, y } ; x €B, y € A ' and x # y } . Definition 7.4.5. The symbol 0 +.AH(a(O), ..., Q(S - 1)) indicates that whenever X is an almost homogeneous subset of o1with tp(X) = up,then there are i with i < s and A in [X]""(') such that [AI2 C_ Ai. The next lemma reduces the problem of proving Theorem 7.4.3 to the problem of constructing almost homogeneous subsets of arbitrarily large (countable) order type. Lemma 7.4.6. For all a there is p such that 0 +.AH(a,

..., a).

Proof. We shall show the following: given ordinals a(O), ..., a(s - 1) such that for all i with i < s and all y with y < a(i) there is p(i, y) such that p(i, y) +. AH(a(O), ..., a(i - l), y, a(i -t l), ..., a(s - l)), then there is such that 0 +.AH(a(O), ...,a(s - 1)). It follows by induction on ao, ..., that for all ao,..., as-l there is /3 such that 0 +AH(ao, ...,&+I); in particular the lemma is true. If any a(i)is zero, the claim above is trivially true (with arbitrary), so suppose that always a(i)2 1. Choose sequences (a(&n); n < w>such that = &,(w"("~); ri < a). Put a(i, 0) G a(i, 1) G ... < a(i) and

p(n) = max @(i,a(i, n)); i < s} and 0 = Z0(fl(n);n < a). We shall show that

0 +.AH(a(O), ...,a(s - 1)) .

(1)

So take X from [wllW' such that X is almost homogeneous. Then there are X o in [X]w'(o), A. in [XIwp and i(0) with i(0)< s such that X o < A0 and X o c A. G Ai(o . Inductively we can contjnue and find X , in [A,-1 ]WP(,) A, in [A,-l]wB and i(n) such that X ,
if n < m, then X ,


and X , 8 X ,

C_

A,(,)

.

(2)

There i s N in such that i(n) is constant for n in N, say with value j. For each n, the choice of p(n) ensures that P(n) +.AH(a(O), ..., a(j - l), a( j , n), a(j + I), ...,a(s - 1)). Thus for each n, either there are i with i # j and C, in

Ordinav partition relations

176

Ch. 1.4

[Xn]wff(i) such that [C,]' C Aj, or there is B, in [Xn] w"(i'") such that [B,,]' C A , . If for any n the first of these alternatives is true this gives i and C in [XIwdd with [CI2 C Ai. On the other hand, if for all n the second alternative holds, consider B where B = U{B,;n EN}. By ( 2 ) , tp(B) = Z;o(wff(i~"); n EN) = a") . From ( 2 ) , B,e B, C_ Aj if { n, m ) E [N]', and also [Bn]' C A . by the choice of B,. Thus [B]' C A,-, so also in this case there is B in [X]udn with [B]' C A,. Thus the relation (1) holds. This completes the proof. We shall later make use of the following lemma to construct large almost homogeneous sets from smaller ones. Lemma 7.4.7. For each n with n < w let X,, be an almost homogeneous subset of wl. Suppose X , < X , whenever m < n, and suppose there are integers i(m)(with i(m) < s ) such that

m < n *X ,

Q

X, C

Then U{X,; n < w } is almost homogeneous. Proof. Take any subset A of U{X,; n < w } with tp(A) = 0". Take any (3 with (3 < a.If A C X,, U ... U X , for some m, then by Theorem 7.1.3 there must be k with k < m such that tp(A n X k ) = wQ.Since X k is almost homogeneous

this gives B in [A fl Xklw' and A' in [ A n X,]"" such that B < A ' and B s A' C A,- for some i. On the other hand, ifA is cofinal in U{X,; n < 0) then tp(A nX,) 2 w p for some m. Then if B E [A n X,]"' and A' = A - ( X , U ... U X,) (so tp(A') = a")it follows that B s A' C A,(,). Thus U{ X,,; n < w } is indeed almost homogeneous.

The construction of almost homogeneous sets will depend on refining pairs ( A , X ) of subsets of w1 (with X uncountable) such that A @I X i s "almost contained" in Ai for some i. We shall write G(i) for the set of pairs ( A , x> where A is in [wllW" for some 01, X is in [wlIHl , A < X and A @ X is "almost contained" in Ai. "Almost contained" is to have the sense that n o matter what subset A' of A of the same order type as A or what uncountable subset X' of X are chosen, for every (3 smaller than a there is a subset A" in [A']"', not cofinal in A', and an uncountable subset X " of X ' such that A"@ X " is "almost contained" in Ai. This leads to the following inductive definition of G(i). The relation ( A , X ) E C(i)where i < s, A E [w1IWLy and X E [allH1 is

Ch. 7.4

177

Partitions of / w 1 / 2

HI = 1) then (A, x> E C(i) if and only ifA < X a n d A 8 X C Ai .

defined by induction on a.If a = 0 (so

If a > 0, supposing the relation (A', X') E G(i) has already been defined whenever tp(A') = upwhere 0< a,then

U,X ) E C(i) if and only ifA < X and whenever A' E [A]"" and X ' E [ X I N 1for , every 0 with 0 < a there are A" in [A']"'

not cofinal in A' and X " in [ X ' I N 1such that (A", X " ) E C(i) .

It is clear from the definition that if ( A , x> E C(i) and X ' E [XI"l , A ' C A with tp(A') = tp(A), then U', X ' ) E G(i).

Lemma 7.4.8. Given A in [wllw"and X in [ul]"'then there are A' in and X' in [XI" such that ( A ' , X ' ) E C(i),forsome i with i < s. Proof. Since H I < No and = N,, we may suppose A < X. If (Y = 0, for i w i t h i < s p u t X i = { x E X ; A s ( x } C A i ) . Forsomeiwehave lXil=H1; then ( A , X i ) E C(i). Now suppose a > 0, and suppose inductively that the lemma is true whenever tp(A) = upwith 0< a. Suppose the lemma is false for particular A0 in [u1IWa and X o in [ u l l Hthus 1;

1x1

vi< s VA' E [

A ~ VX' ~ wE ~[ X ~ I " ( ( A ' ,x')4

,

c(i)).

(1)

Inductively define chains A . 2 A 2 ... 2 A , from [Aolw" and X o 2 X1 2 ... I X , from [ X O ] " so ~ , by (1) ( A , X i ) 4 C ( j )wheneverj< s, as follows. Since in FAj, Xi) 4 C ( j ) ,by the definition of C( j ) there are Aj+, in [Ailwa, [Xi]" and P ( j ) < a such that VA" E [~i+,lw") VX" E [ X ~ + ~(A" I " not ~ cofinal in

Ai+l =$(A",X " ) 4 G ( j ) ) .

(2)

Let 0 = max { b ( j ) ; j < s}. Choose B from [A,lwP not cofinal in A,. By the inductive hypothesis applied to B and X,, there are B' in [B]"' and X' in [X,]" such that (B', x'>E C ( j ) for some j. If 0(j) = 0,since X' C X , C and B' C B C A , C Aj+, with B' not cofinal in A . + l , this contradicts (2). If /3(1) < 0 by definition of C ( j ) there are B" in [Bl]w'o.) not cofinal in B' and X " in [X']" such that (B",X " ) E G ( j ) . Now X " C Xi+,, B" C Aj+l and B" is not cofinal in Aj+l so again ( 2 ) is violated. In either event a contradiction has been reached, and thus the lemma cannot be false for A0 and X O .This completes the proof.

178

Ordinary partition relations

Ch. 1.4

Lemma 7.4.9. Let ( A , X ) E C(i). Then there are x in Xand a subset A' of A with tp(A') = tp(A) such that A ' s { x } C Ai. Proof. Let A and X be given with ( A , x> E C(i), and suppose tp(A) = wa. If a = 0, the result in trivial, so suppose a > 0. Make the inductive assumption that the lemma is true for all pairs (Ao,Xo>where tp(A0) < tp(A). For a in A and p with p < a define ~ ( a p), = {x EX;

3 B E [ AIW'

(a < B, B is not cofinal in A ,

and B o {x}L A,)} . P u t X * = X - U{X(a,/3);aEA a n d p < a } . We shall show that X* # C. Suppose on the contrary that X* = @.Then IX(a,/3)1= H1 f o r s o m e a a n d p . L e t A ' = { a f E A ; a < a ' } , t h e n i t follows from Theorem 7.1.3 that tp(A = wa. Since ( A , X ) E G(i), there must be X" and A" in [A']'' not cofinal in A' such that A"@ X " E G(i). in [X(a, Thus by the inductive hypothesis applied to A" and X", there are x in X" and B in [A"]"' such that B s {x} C Ai. However then a < B and B is not cofinal in A , so x 4 X(u, p). This contradicts x EX" C_ X(u, 0). Hence X* # Now choose x EX*. Take ordinals a(n) where n < w such that always a(n) < a and wa = Co(w(u(n); n < w). Then x 4 X(a, a n ) for all a in A and all n. Thus we can choose inductively sets B, in [A]WJn1but not cofinal in A , for n with n < w , so that sup B,-l < B, and B, s {x}C Ai. Put A' = U{B,; n < a};then A' E [A]Waand A ' s {x} C Ai. This proves the lemma.

f$

The next lemma will serve as the starting point for an inductive construction of an almost homogeneous set.

Lemma 7.4.10. For n with n < w let (A,, x> E G(in).Then there are x in X and subsets A ; o f A , with tp(AL) = tp(A,) such that A ; s (x} C Ain for each n. Proof. Suppose given X and the A , such that (A,,,X) E G(i,). Put X, = {x EX; 3 -4; C An(tp(A L) = tp(An) and A ; 8 { X} C Ai,)}. If X = U{ X,;n < O} then there is some X n with IX,l= N l . Thus (A,, X,) E G(i,), s o b y Lemma 7.4.9 there is x in X, and A' contained in A , with tp(A') = tp(A,) such that A ; s {XIC Ai,; this contradictsx EX,. T h u s X # U{X,; n < a}, and so the lemma holds..

For the remainder of this section, for each non-zero fix upon a sequence of ordinals (a(n);n < w ) such that always a(n) < a(n + 1) < a and wQ=

Ch. 1.4

Partitions o f l w l l 2

179

Co(w"("); n < b).We shall eventually construct almost homogeneous sets of order type upfor arbitrarily large p by induction on p. We shall do this by finding for each n almost homogeneous sets D, with tp(D,) = up(')such that there are subsets B, in [D,]wp(n)which fit together to give an almost homogeneous set of type up.In order to d o this, we shall need a way of choosing, w times, elements in each D, in such a way that the set of those elements which are chosen at every step constitutes a subset of D, of type

,P(n).

We now look at a method of making such choices. In general, let a set A in [w1lw" be given. Let also a finite family d of subsets of w1 each of order type an ordinal power of w be given. For each integer k, define operations Sk on such sets A and such finite families d as follows: if a = 0, then SAA) = { { A } } ,if a > 0, then S&)=

w"(k)

{{B,C};BE [A]

, C E [AIWaandB
if d = {Ao, ...A,} then S k ( d ) = {{Bo, CO,..., B,, C,}; {Bi, Ci} ESk(Ai) for i = 0, ...,rn} with the understanding that if tp(Ai) = w0 then Bi= Ci= Ai.Thus { A } ) = S&). If d is a finite pairwise disjoint family then so is any member of S k ( d ) . GivenA from [w1Iwa, suppose we take a sequence of families ( d , ; n < w ) such that do = { A } and there is some integer k so that always

Sk(

E S k + n ( d n)

when n 2 0 .

Thus d provides two disjoint subsets of A , one small, one large; d z provides similarly two subsets of each of'these; and so on. If we put A , = ud,, then A n is to be the set of elements o f A chosen at step n. Thus'n{A,;n
Lemma 7.4.1 1. In the above situation,

n { A , ; n < o}E [AIw".

Proof. By induction on a,where tp(A) = w". If a = 0 then for every k and n we have Sk+,(A) = { { A } } so n{A,; n < w } = A E [AIWO. SO now suppose that a > 0 and make the inductive assumption that for any p with < a,given n < w ) where CM 0 = { B } and for all n, gn+l E Sl+,(?3,) a sequence (qn; for some finite I , where tp(B) = op,then tp( n{ UCM n;n < w } ) = up, Now take the sequence (A,; n < w>described above, w i t h d = { A } where tp(A) = wa. Inductively choose members B , and C, o f d n+l with tp(C,)=w" as follows. Since d , E S k ( A ) there are Bo in [A]Wdk) and COin [A]w"such

180

Ordinary partition relations

Ch. 7.4

since d , + 2 E that Bo < CO and {Bo, CO)G d1.Supposing C, E there are subsets B,+l of C, with tp(Bn+l) = oa(k+n+l) and tp(Cn+l) = we such that B,+1 < Cn+l and {Bn+l, Cn+l} C d n + 2 - Thus always Bn < Bn+l. Consider now the set Bo. There is one subset of Bo in d1,namely Bo itself. In 942 there are two subsets of Bo and together they give a member, say g g, of Sk+l((X 0). In J Q j , these get divided t o give a member of Sk+2(Ca g), and so on. In general, if we put cx3 on = { X E d n + 1 ; X C Bo} then Sk+n+l(dn+l)

~ O O ={ B o ) ;i f n

2 0 then%,+l

ES(k+l)+n(qOn).

by the inductive hypothesis, if

Since tp(B0) =

then Bb E [Bo]Wa(k). We can repeat this with B1, B2, ... . For any m , put ' 3 m , = { X E s Q n + m + l ; X C B m ) . Then g m n =

{Bm 1;Bmn+1

ES(k+m+l)+n(cIomn)

.

So by the inductive hypothesis, if

BL = n{Uq,,;n < 01

then BL E [Bm]Wa(k+m). Now for any n we have

BL

C_

U g m , C Udm+n+1 -Am+n+i ,

and if 1 < m then

BL CB, CCm-l C-Ci-1 C U d , = A , . Hence always BL C fl{ A , ;1 < a}.Thus if B = U { B L ;m < w } then B C { A , ; 1 < o}and tp(B) = Co(oa(k+m):m < o)= we. This proves Lemma 7.4.1 1.

Lemma 7.4.12. Take any finite painvise disjoint f a m i l y d and any uncountable set X such that for all A in 94 there is iA with iA < s such that ( A , X ) E G(iA). Then for any integer k there are a f a m i l y 9 in S k ( d ) and a set X ' in [XI* 1 such that for all B inc1s , B C A E d =*(B,X')EG(iA). Proof. Write d = { A o , ..., A m } . Suppose tp(A,) = oaiand ( A j , X ) E G(ij). Take subsets A; and AT of A , with A; < A,? such that tp(A;) = oe,(k) and tp(A *) = we, (but if a, = 0, put A; =A,? = A , ) . Choose sets X i from [XIK

Ch. 7.4

Partitions of / w 1 J 2

181

for j with j < m + 1 so that X o = X and Xj+l is in [X.]' such that there is A for which ( A j , Xj+l)E G(ij).This a subset A; ofA; with tp(A;) = uaj(k) choice is ossible since ( A j , X ) E G(ij). Put X ' = fl{ X i ;j < m + 1) so X ' E [XIR1and (A;, X ' ) E C(ij) for each j. Also ( A T , X ' ) E C(i.) since i' tp(Af) = tp(Aj). Now {A;, A T } E S k ( A j ) so if we put CM = { A d , A ; , ..., A;, A : } then CZJ E S k ( d ) and 'I3 with X' has the required property. In the next lemma we shall finally construct the almost homogeneous set of order type upfor arbitrary 0. The more complicated statement in the lemma is needed for the inductive construction.

Lemma 7.4.13. Let be given. Take any set X in [ul]' and any finite pairwise disjoint family d = ( A o , ..., A k } with ( A j , x ) € C(ij)(wherej = 0 , ...,k). Then there are B in [XIwpand subsets A; of A j with tp(Ai) = tp(Aj) such that B is almost homogeneous and A; @ B C A,, (wherej = 0 , ...,k). Proof. The plan of the proof is this. We shall construct the almost homogeneous set B by finding almost homogeneous sets B, with tp(B,) = up(,)and B, < B , whenever m < n such that there is i(m) with i(m) < s for which Bm @ B, C A,(,) whenever m < n ; by Lemma 7.4.7 this will suffice. At stage n in the contruction we shall introduce a set D , of order type up(n) from which the elements of B, will be drawn. When finding D,, we shall choose elements of all the earlier D, and of the Aj, say in sets D& and AT, such that D& @ Dn C A,(,) and A T @ D , C Aij. Those elements that are chosen at every stage will form the B , and A;. The proof is by induction on p. The case p = 0 is given by Lemma 7.4.10. So suppose that p > 0, and make the inductive assumption that the statement of the lemma is true for any y with y < 0. Put d o= d and X o = X. By Lemma 7.4.12 we have a familye 0 in S o ( d 0 ) and a set XA in [XOlN such that for all C in C if C C A j then (C,X;) E C(ij).Since p(0) < p by the inductive hypothesis applied to eo and XA there are an almost homogeneous set Db in [XA]wp(0)and subsets C'of C for C in C? with tp(C') = tp(C) such that C'@ Db C Ai. for that i j for which (C,XA) E Gi.. Note that also { C';C E- C,} E S o ( d o ) .by Lemma 7.4.8 there are Do in [ D i ] w p ( oand ) XI in [X,]' such that (Do, X1)E G(i(0))for some i(0)with i(0)< s. Then Do is also almost homogeneous. Put d l = {C'; CE'ZJ o } U { D o } . Since Do C Xo and necessarily A , < XO for each j , it follows that d is a finite pairwise disjoint family. And further, for each A in d l there is i(A) such that ( A , X1)E C(i(A)).Thus we can repeat the process. This leads to the following inductive construction of finite pairwise disjoint

182

Ordinav partition relations

Ch. 7.4

familiesd , uncountable sets X,, and almost homogeneous sets D, in [X,]"'p(n)! Suppose that 4,and X , have already been constructed:'such that for all A i n d ,there is i(A) with i(A) < s for which ( A , X,) E C(i A ) . By Lemma 7.4.12 there are a family C, in S,(A,) and a set XA in [X,] L l ) such that for each C in C, if C C A with A E d ,then (C, X i ) € G(i(A)). Noting that p(n) < 0, by the inductive hypothesis a plied to en and XA , we can find an almost homogeneous set DL in [Xi]Wp and subsets C' of each C in C ,with tp(C') = tp(C) and C'e DL C A i c ~ for ) that A in d ,where C' C C C A . We mention that { C';C E en}E S n ( 4 , ) . Using Lemma 7.4.8 gives Dn in [DL]"p(n' and Xn+, in [XnlN such that (D,,X,+,)€ C(i(n)) for some i(n). Then D,, being a subset of D;,is almost homogeneous. Also C' < D, for every C', since C' C C, D, C X , and C< X , (for each C in en). Put -d,+, = { C ' ;C E en}U {D,}, so d,+,is a finite pairwise disjoint family. And for each A in d , + l ,there is i(A)such that ( A , X,+,) E G(i(A)). Now take any one of the original sets Aj (so j = 0, ..., k). In each of the families 9'2, consider those members which are subsets ofAj. As in Lemma 7.4.1 1, this gives'a sequence of subfamilies ( d i nn;< w ) where

8)

Ajo =

{Aj};Ajn+l E S n ( d j n ) .

Hence by Lemma 7.4.1 1, if A; = n{U d j , ; n < w } then A; C_ A j and tp(AJ) = tP(Aj). d ,consider the subsets of D,. Take any D, (where m < a),and in each s There are n o n e i n d o , ..., dm; in d,+1 there is D, alone; in .dm+2 there are two giving a member of S,+2(D,); in Am+3 these are further divided, and so on. Thus there is a sequence (Q;, n < w ) with (D , ,Cd such that am0 =

{Dm};g r n n + l

ES(rn+l)+n(Gmn).

Thus by Lemma 7.4.1 1, ifB, = n{UCbmn;n < w } then B, C D, and tp(B,) = tp(D,) = Then each B, is an almost homogeneous subset of X. And if m < n then B, < B, since B, C D,, B, C D, C_ X , and D, < X,. Further, if m < n then B, 8 B, C A,(,) since D, 8 X,+, C_ A,(,,,) by choice of D, and Xm+l, and B, C D,, Xn C X,+l. Put B = U{B,; m < a}.By Lemma 7.4.7, B is (~ <)w; ) = up. almost homogeneous. And tp(B) = Z O ( ~ ~ m Finally; we show I ; @ B C Ati if j = 0, ..., k. Given A j , for any integer n let dj,= { A E d , ; A C A j } so U -4, gives the set of members of A j chosen at the n-th step. An easy induction on n shows that ( A , XA) E C(ij) for every A in dj,. Then the choice of DL and d +,, ensures that A t e Dh C A,, for every A' in djn+,, so ( U d j , + l ) e DL C Aij. Since A; C_ UoQj,+, and

Ch.1.4

Partitions of [ w 1 ] 2

183

Bn C Dh,thus A;@ B, C Aii; hence A;@ B C Aii. This completes the proof of the lemma. Lemma 7.4.13 contains the last step necessary for the proof of Theorem 7.4.3.

Proof of Theorem 7.4.3. Let the countable ordinal a be given, and take the partition [w1I2 = A, U ... U As. By Lemma 7.4.6 there is P such that P+AH(a, ..., a). From Lemma 7.4.13 there i s B in [ w l ] w psuch that B is almost homogeneous for this partition. So by Definition 7.4.5 of the symbol P+AH(a, ..., a),there isA in [BIw" such thatA is homogeneous for the partition. Hence there is surely a subset of w1 of order type a homogeneous for the partition. Thus indeed w1 + (a),".