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Chapter IV Trees and Positive Ordinary Partition Relations
CHAPTER IV
TREES AND POSITIVE ORDINARY PARTITION RELATIONS
A general framework for tree arguments is presented, and this is applied in order to derive the main results of the form K + ( & ) : < s , where K is a regular cardinal upon which no “large cardinal assumptions” (see Chapter VII) are imposed. 13. TREES
We defined trees in Section 10, where we needed the concept for the second proof of Ramsey’s theorem, but, for the sake of completeness, we recall the definition here : A tree is a partial order (7; <) such that, for any x E 7; the set of its predecessors, pr (x)=pr (x, (T, <))={y E T : y < x ) is wellordered by < (we assume that < is irreflexive, it., that 7x < x holds for any x E 7;though this is not an important assumption). We sometimes write T instead of (7; <).For any x E T, the order o(x)= o(x, (IT; <)) of the element x is the order type of pr (x). For any ordinal a, the set T[a] = (x E T: o(x)= a ) is called the ath level of T, and Tla={x~ T: o(x)
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TREES
THEOREM 13.1. Let (7: <) be a tree, and assume that K I IT1 is regular. Let I be a cardinal with w s i. < K . Then T has a path of length 2. 4 1 provided one of the fbllowing two conditions is satisfied. a) I Is @)I < K holdsjor every path p of T, and < K,for every K, < K and i., < j.. b) There is a cardinal CT with a?< iisuch that I Is ( p ) (5 d[P1""1 holdsfor eoery path p of T. (Jf A>w, rhen one cun write CTIPI" instead Of'&"""1 here.)
PROOF. We have to prove that T has length 2 j. 4 1, as this is clearly equivalent to the existence of a path of length j.4 1; i.e., we have to show that TI;.# T As T was assumed to have cardinality 2ii, it is enough to show to this end that ITILI
As I < K and K is regular, it will be enough to show that 1 TI a [< K holds for any
a o ) , the range of each ,fp is included in do.Hence the number of the fis is I (&)l'ls~~?
contradicts our choice of q, establishing b). (The remark in parentheses can be proved similarly; or, alternatively, it is a direct consequence of the main result if we add o new levels at the bottom of T.) The proof is complete. Another result establishing the existence of long branches is THEOREM 13.2. Let K, I , p, and t be cardinals, and suppose that K is regular, (0< )T < p < I , and d p < K. Let 9 be an ordinal, and let T be a set offunctions f E "r, and assume that if f e T , then f " a E T for any a s 3. T ordered by
0I A,
u
as9
6 Combinatonal
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TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
inclusion is a tree. Assume that 1 TI 2 ti and that
I (tE dom ( f )f(O>o)l : holds for
any f~ 7: Then T has
Q
path
(1)
of length 1 i1.
PROOF. It is obvious that T is a tree. As before, it will again be sufficient to prove that l’Tlj.]
The number of functions f
E
U ‘7
satisfying ( 1 ) is I
a
i.e., l T l q [ < ~which , we wanted to prove.
14. TREE ARGUMENTS
One is often interested in constructing a ‘long’ sequence ( x a :a < 8 ) ofelements of a set S that has ‘nice’ properties. For example, in the proof of Ramsey’s theorem we were given a coloring f :[S]‘+k, and we constructed a sequence ( x i : i < w ) in such a way that we had f ( u u { x , ) ) = f ( ~ u ( x , whenever ~) m l n < w and u ~ [ ( x ~ : i < m ] ] ‘The - ~ . procedure we followed In the second proof of Ramsey’s theorem was that, when we had already constructed a finite sequence { x , : i < l ) that had the above property (with m s n < l ) , then we continued this sequence in several ways that, in a sense, represented all possible continuations, thus ensuring that at least one of the sequences had an infinite continuation. In constructing these sequences, it was helpful to define the set S((x, : i < I ) ) of the potential successors of the sequence ( x i : i < 1). The situation in Ramsey’s theorem was simplified by the fact that we did not have to consider sequences of transfinite length. The proof of Theorem 11.3 above was in many ways analogous, though an additional complication was caused by the fact that the elements of the ‘long sequence’ to be constructed were subsets, rather than elements, of the underlying set. We are now going to consider a general framework for arguments of this type, called tree arguments, and we shall prove a few results concerning the sizes of ‘nice’ sequences that can be obtained in this way.
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TREE ARGUMENTS
The underlying concept of tree arguments is that of a partition tree, also called ram$fication system: DEFlNi-rioN 14.1. A partition tree is a pair ( E , S ) , where E is a set (called the underlying set), and S is a function with the following properties:
(i) ra ( S ) s Y ( E ) ;(ii) dom ( S ) =
u<>
a
' p for some ordinals 9 and p; (iii) S(O)=E
(notethatOE'p= ( O ) ) ; ( i v ) i f J ; g ~ d o(mS ) a n d j s g , t h e n S ( g ) s S ( f ) ; a n d(v)if f ~dom (S), and q=dom ( f )is a limit ordinal, then S ( f ) =
n S(,f ' a ) .
a
S will usually be defined by transfinite recursion with the aid of the recursion for mu la
for any ordinal 5. Here G is a given function whose range is a subset of Y( E ) and whose domain is what it ought to be according to (1). In order for (1) to define a partition tree, G has to fulfil the following conditions with X = Sldom ( f ) : G(O,O)= E ,
(iii')
if q =dom (f )is a limit ordinal, so that (iii), (iv), and (v) of the above definition be satisfied. If these conditions are satisfied, then (1) obviously defines a partition tree. (iii') and (v') mean that we have no freedom at all in specifying G ( f ,X )unless dom ( f ) is an ordinal of form a $ 1 for some a; that is, in order to define a partition tree, we have to specify G(f; X ) only in case Jom ( f ) is a successor ordinal. We are going to define some auxiliary concepts concerning a partition tree ( E , S). Noting that c always denotes strict inclusion here, for any f ~ d o m( S ) write (3 1 N f ) =S(f)\ {sk):fCgE dom 6))
u
and, moreover, put
T = { f E d o m ( S ) :S ( f ) # O )
In view of (iv) of the above definition, Tordered by inclusion c is a tree (this 6'
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TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
justifies the word tree in the term partition tree), and we have R(J')=S(,j')\U(S(g):,f~g~T&dom(g)=dom(f')/l).
Our main interest below will be how long the branches of T are, as a branch of T will define a 'nice' sequence of the same length. We shall occasionally refer to a partition tree as a quadruple (E, S, R, T), where R and T are defined from S in the way described above. We now give some explanation about the intuitive meanings of the letters E, S, R, and 7: In the simplest case, we want to construct a long 'nice' sequence (xu: a < q) of pairwise distinct elements of E. We construct this sequence step by step, making alternative choices at each step, the number of alternatives being Limited by IpI, where p is the ordinal in (ii) of the above definition. Having constructed a sequence ( x ~ c(: < 5 ) of length t, a function .fi 5-p keeps track of the alternative choices made, and the sequence (xg:a < t )is obtained in the form ( s ( f " a ) :a < ( ) , where s is a function from T into E defined by transfinite recursion (simultaneously, or subordinated to, the recursive definition of S as T one described in (1)) in such a way that s ( f ) E S ( , f ) holds for any , ~ E Usually, can pick an arbitrary element of S ( f ) as s ( f ) , that is, S ( , f ) is the set of potential successors of the sequence ( s ( f " a ) : u)), where Y < P . Using the results of the preceding section, we are going to prove a theorem on the length of branches in a partition tree which will be one of our main tools in constructing large homogeneous sets. We first need a lemma, which simply says that each element of E is thrown away (or used up) at some'stage:
LEMMA 14.2. For any partition tree ( E , S, R, T ) we have E = ( j { R ( f ) :f e T : .
85
T R t E AR(illMF.NTS
PROOF. For an arbitrary x E E put T , = { y E T:xES(y)}.
This set is not empty in view of (iii) in Definition 14.1, and (T,, c ) is obviously a tree by (iv) there. By Hausdorffs Maximal Chain Theorem (see Subsection 4.7), there is a branch b of this tree; put f = b. We claim that , f ' ~h. This is obvious if b has a last element in the ordering c ,as this last element can only be ,f.Hence, assuming that ,f$ h, 5 = tp ( h , c ) = dom (.f ) must be a limit ordinal (we cannot have <=O, since h is not empty, as Tv #O). Then
u
xE
( q g ) :g E b; =
n (s(,jh a ) :a < < ) =s(f'),
where the last equality holds in view of (v) in Definition 14.1; observe that (ii) there is also used to show , f dom ~ (S)(this is why it is an important technicality ,'u ). So . f ~T,; therefore
to take dom (S)= (J "p instead of dom (S)= US 9
f$ b
U
means that b u {f). is a chain of T, properly including b which contradicts the assumption that b is a branch. This establishes the claim f E b c T,. Hence x E S(f ). I f g is an element of dom (S)that properly includes f then x $ S k ) , since otherwise b u {g)would be achain of T, that properly includes b which would be a contradiction. Hence x E R ( , f )holds by the definition of R in (3)above. As x E E was arbitrary, this completes the proof. The main theorem of this section is simply a reformulation of the results of the preceding section in the light of the lemma just proved. We recall that ims (S, T) denotes the set of immediate successors in T of the element f: In the present case we have ims (,A T ) = { g E T :fcg eL dom (g) =dom (f) i 1 3. THEOREM 14.3. Let ( E , S , R, T ) be a partition tree; let K be a regular cardinal with K 5 1 E I, and assume that IR (f) 1 w, then one can write 0(1'"+ I)' instead of d[IfI+I1'"1here.) c) There are cardinals p and T with ( O < ) . r < p < i l and ~ P < K , we have T s 'r,forsome ordinal 9, and I { t E dom (f): f (<)> 0 )I
u
asY \
PROOF. I R ( f ) l < K implies I TI 2 K by virtue of the preceding lemma. Hence the result is a simple restatement of Theorems 13.1 and 13.2. Note that each path of limit length has at most one successor in case of a partition tree (E, S, R, T ) .
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TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
Hence, in cases a ) and b), the inequalities for 11s ( p ) l required in Theorem 13.1 follow from the analogous inequalities for lims ($)I here. There is a possibility of generalizing the concept of partition trees to cases when S (,f) assumes values in any partially ordered set instead of the power set of a given set E . A case in point is e.g. the following simple result observed by K. Kunen: given a cardinal K, any (K', 3)-distributive complete Boolean algebra B that satisfies the K-chain condition is atomic. To prove this result, one has t o use partition trees where S ( f ) E B. (For the definitions of the concepts here, see e.g. Sikorski [1960]; (K', 3)-distributivity here is t o be understood in the sense of ( < K', < 3)-distributivity, and not of I( K', I 3)-distributivity.)
15. END-HOMOGENEOUS SETS Let a be an ordinal and T a cardinal, and let ,fi [a] <"-+T be a coloring. Call a set X c a end-homogeneous with respect to ,f iffor every finite set u c X and for every p, v E X with max u < p, Y. we have f ( U U
(p))=.f(uu
{\I]).
(11
I f f is only a partial coloring of [u] < w i.e., iff is a function from a subset of [a] < w , then call X E end-homogeneous ~ with respect t o .f if ( 1 ) holds whenever u E [XI <", max u < p , v , and u u { p i , u u E dom ( , f ) . The partition relation a+
(8)r
(2)
means the following: for every coloring ,$ [a]""+r there is an endhomogeneous set of order type p. End-homogeneous sets are very important in establishing positive partition relations; in fact, they were already used in both proofs of Ramsey's theorem in Section 10. The following simple lemma throws some light upon the situation. LEMMA 15.1. Let r 2 1 be un integer, 7 u cardinul, ond let a, b, 1,: (( < T )be ordinuls. Assume p+(\a:);:;. (3 1 Then (r -+(p), implies
and a+ ( p 4 1), implies
(r-+(\j:):.
u+(\,:+
1
I,;.<,.
(4) (5)
An instance of relation (4) was used above in the proof of Ramsey's theorem, and relation (5) will be uskd below in the proof of the Stepping-up Lemma (Lemma 16.1).
87
END-HOMOGENEOUS SETS
PROOF. Let ,/’:[ c r ] * - t ~ be a coloring. First we verify (5). Assume to this end that cr+(fiit), holds; then there is a set X of order type p i 1 that is end-homogeneous with respect to .f: Let i be the maximal element of X, write X ’ = X \ !(),and define the coloring g: [X’]*-’+T as follows: put g ( u ) = = , f ( u u (0 )for every u E [X’]‘- ’. By ( 1 ), there is a < T and a set Y E X’of order type 11,. such that Y is homogeneous of color 5 with respect tog. i.e., g”[ Y ] ‘ - ’ = = (0.But then we have
<
, f ( i ? ) = , f ( ( r \ (max
vI)u(0)
for any I’ E [ Y u (<;]‘by the end-homogeneity of X, and the right-hand side here equalsg(z-\ (max I . ) ) = by < the definition ofg. Hence Y u (i) is a homogeneous set of order type v , . i 1 and of color with respect to ,I; which proves (5). In the proof of (4), assume first that fl is a limit ordinal. The relation ~ c - - + ( f i ) ~ implies the existence of an end-homogeneous set X 5 CI of order type with respect t o the coloring ,J [cr]‘-*r. Given an arbitrary u E [XI‘-’, put g ( u ) = = , f ’ ( u u ( v i ) for a V E Xwith v>max ( u ) ; there is such a 11, since t p X = p was assumed t o be a limit ordinal, and g ( u )does not depend on the choice of vin view ofthe end-homogeneity of X. By (3). there is a 5 < T and a set Y G X of order type Y: that is homogeneous of color 5 with respect t o the coloring g : [XIr-’+?: clearly, Y is then also homogeneous ofcolor with respect to .I:This establishes (4) in case p is a limit ordinal. If B=p i1 for some /?’,then we have t o work with the last element of X as we did in the proof of (5). It is, however, simpler t o observe that (3) implies
<
<
1,r;f
P’+(v,
in this case (indeed, by taking away the last element of p, we can take away only the last element of a homogeneous set), and so we have cr+
( ( I #
1) i 1 );
<
~
in view of ( 5 ) with p’ replacing p. The proof is complete. The following lemma, due t o ErdBs and Rado, confirms the existence of large end-homogeneous sets. It will usually be applied together with the preceding lemma t o prove the existence of large homogeneous sets. I t is, however, also important in itself: in Section 45 we shall give some applications of it concerning set mappings in cases where the corresponding partition relations would lead to weaker results. LEMMA 15.2. For any cardinals
K 2w
and i L 2 Mie hare
(E.F)+--t(K/
l)i.
(6)
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TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
PROOF. Writing E = (AF)+, let f : [ E l <"--+A be a coloring. We are going to define a partition tree ( E , S, R, T ) , called the canonical parfitionfreeassociated with the coloring .f: To this end, write dom(S)=
u
(7 1
'p,
US9
where p and 9 will be specified later. As we mentioned in the preceding section, we can define S by the recursion formula
k)),
S(g)=G(g,
where G has to be specified only in case dom (g) is a successor ordinal and S(g * 5 ) # 0 for any 5
5 E S(g')\
(9)
/?
PP):
{~(g')',, define the coloring
,fg,.t: [E,.]<"-A
&.& 1=f (u u { t 1) for any u E [E,.]
<"'.
For any two
by putting (10)
5, q E S(g')\ {~(g')),put
E ~is .obviously an equivalence relation. The number of equivalence classes under is I the number of possibilities for the coloring &.,r, i.e., is at most =g6
where we recall that a = d o m (g'). Let S,.,,, v < p , be an enumeration of the Here . p (cf. (7)) has to be large enough so that we equivalence classes under E ~ . may enumerate all the equivalence classes (as these classes are pairwise disjoint, p = E = I El clearly suffices); if I p Jis larger than the number of these equivalence classes, then put S,,,=O as many times as necessary. Write
S(d =&@ ,)
f
and, for further reference, if S(g)#O, then write fg
=f,,.<
9
END-HOMOGENEOUS SETS
89
where 5 E S k ) ; clearly, f, does not depend on the particular choice of 5 here. By (lo), we have fe(u)=f(uu
{tl)
(14)
for any finite set u s {s(g * P ) : j < d o m (g)) and any ~ E S C ~This ) . finishes the definition of S. Choose 3 so large that S ( g ) = O for any g ~ ' p clearly, ; 9= E + suffices since, for any g with S ( g ) # O , the set {s(g p a ) : a s d o m (g))
consists of pairwise distinct elements of the cardinal E. Putting, as usual, T = { h Edom (S): S(h)#O). , we clearly have R ( h ) = { s ( h ) )for any h E 7; i.e., we have IR(h)l= 1
(15)
for any h E 7: Taking h =g' in (12), we can conclude that lims ( h ) l s AI[lhl+'l''"l
(16)
holds for any h E 7;where ims (h) denotes the set of immediate successors of h in T; in fact, exactly one immediate successor of h in T corresponds to each (nonempty) equivalence class under -,,. Noting that 1 El = E = (,IF)+, we can conclude from Theorem 14.3.b with (A!)', K, and 1replacing K , I, and a, respectively, that T has a path of length K + 1, i.e., that there is a g E T with dom (g)= K. Write 5,=s(g . first that for each M ~ KObserve (8), we obtain that
pa)
t,
5, = min S(g + a )< tP provided a <
K,
since 5pE
Sk * P I G S k " a ) \ {t,)
holds by our construction above. Hence the set X = { 5,: 1Ia s K ) . has order type K / 1. We claim that X is an end-homogeneous set with respect to f. In fact, if u E [XI and max u = 5, < t p tY , (or a= 0 and t P , E X in case u is empty), then we have a 4 1s P, 7, and so to,tYE S(g (a 41)). Hence (14) implies that
ry
f ( u u { 5 P ~ ) = f , ~ ( , ~ l , ( u ) = f (it),;) uu holds; this shows that X is indeed end-homogeneous. The proof is complete.
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TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
COROLLARY 15.3. Let
K
and z be cardinals with z < K and
Iv
K > (11.
Then
iI),
(2")++ ( K + and
I),
(2")++
hold.
PROOF. For (17), replace E. and K with 2" and K + , and, for (18), with z and K , respectively. (18) will be used for the proof of the Stepping-up Lemmas in the next section; (19) will be used in connection with set mappings (see Section 45), and not in order to establish a partition relation. 16. THE STEPPING-UP LEMMA
The following lemma, one of our most important tools in establishing positive ordinary partition relations, is an easy consequence of the results of the preceding section:
LEMMA16.1 (Stepping-up Lemma). Let integer, and v: ( 5
~
and W
5
be cardinals, r 1 2 an
(V:);f
ti+
holds, then we haue
K
(2"++(vci l)?<,
PROOF. We may assume that v: 2 r holds for all t < z (cf. Subsection 9.5). In this case, noting that K is infinite and r r 2 , (1) implies that T < K ; hence, we have (2"+
+(ti
i1 ),
according to (15.18) in Corollary 15.3. Hence (2) follows from (15.5) in Lemma 15.1 with (2!))+ and K replacing 01 and 8, respectively. The proof is complete. This result alone has nontrivial corollaries. As an illustration, observe that the relation
(Po)+
+
(K,I;,
which is a theorem of Erdos and Rado, follows from the trivial relation Nl+(N,)A. This is, however, not the appropriate place to go into discussing the corollaries of the above lemma, since as we shall soon see, stronger results can be obtained in the case r = 2 by more refined methods. This will be done in the next section.
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THE S T E P P I N G U P L E M M A
The aim of the rest of this section is t o derive a result saying that certain partition relations hold simultaneously; this result will have an importan1 application in the proof of the General Canonization Lemma, given in Section 28. We start with the following
DEFINITION 16.2. Let CI and p be ordinals, N a set of integers 2 1, and cardinals. Then the partition symbol
'si
~ (Ei N
11
called the simultaneous ordinury purtition symbol and read as " a arrows simultaneously for i E N with.ri colors", is said to hold if we have the following: Given a n arbitrary coloring ,h: [ c x ] ~ + T ; for each i E N , there is a set X c CI of order type p that is homogeneous with respect to each ,fi, i E N , i.e., for any i E N there is a { < r , with .f';'[Xli= it;.T h e negation of the above symbol will, as usual, be written as ay(p):;" . We mention a few natural variations of the above symbol. Where k and 1 are integers, the relatidn
CIy (/jf,< means the same thing as
as2(pi; I j
I
<1
~ S I < I ~
The meaning of the symbol
$3( pf,,, . ..k" ., IA".
will be defined as
Note also that
.SI:m(fi):'
,
--.I!..
Ik". . . .k"
I;
ol"z(p):F"' means the same thing as a-(fi )Zw (cf. Subsection 8.6).
Using the ideas ofthe proof of Lemma 15.1, it is easy to establish the following:
16.3. Let r( 2 1) be an integer, LEMMA cardinals. Assume that
-(Pi holds with ~ = m a x{ ' s i : 2 < i < r ) . Then
CI,
b, and
I>,
v ordinals, and zi ( 2 5 ; i S r )
(3)
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TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
implies
Iv
aszlf+m(v+ 1 f,si
PROOF. For each i with 2 < i < r , let ,h:[ali+7, be a coloring. Put , f =
u ,f;;
Z i t r
then ,f maps a subset of [a] into 7 = maxZsicr7i.By (3), there is a set X E a of order type 4 1 that is end-homogeneous with respect to f: Write X = X'u{C}, where 5 is the maximal element of X. For each integer i with 2 < i 1 r define the coloringg,: [X']'-'+rbyputtingg,(u)=fi(uu (5))foranyu E [X']'-'.Then, by (4), thereis a set Y E X of order type v that is homogeneous with respect to each g i , 2 5 i s r . We can now conclude in exactly the same way as we did in the proof of (15.5) in Lemma 15.1that Y u { [). is a homogeneous set of order type v i1 with respect to each coloring ,h, 2 <_ i 5 r. The proof is complete. We can now prove the following:
LEMMA 16.4 (Simultaneous Stepping-up Lemma). Let r bean integer, t i ~ and w T, 2 1 (2 5 i 5 r ) cardinals, and v an ordinal. Assume that
holds. Then we have (2"+sz(,&
1);,-*
PROOF. We may assume r 2 2 and v 2 r here (in fact, if e.g. 2 1 v < r, then we can replacer with r ' = v in ( 7 ) ) . Then (6) implies that T ~ < Kfor each 7 with 2 1 i 1 r ; hence we have ( 2 " ) + + ( K / l), with ~ = m a x ~ ~according , ~ , . t ~ to (15.18). Using now the preceding lemma with a= (25))+ and p= K, we can therefore see that (6) indeed implies (7). The proof is complete. Define the operation exp by induction as follows: expo ( K ) = K
and
exp,,, (K)=exp,, ( 2 " ) ,
where K is a cardinal and n'runs over integers. The result we are aiming at (the one we shall need in the proof of the General Canonization Lemma in Section 2 7 ) is the following:
COROLLARY 16.5. Let r 2 1 be an integer and
K
~
aOcardinal. Then we have
MAIN RESULTS I N CASE r = 2 AND K IS REGULAR; COROLLARIES FOR
r23
93
PROOF. For r = 1 the assertion says K + - - + ( K + ) ~ ,which is obviously true. If (8) holds with some r.2 1, then the preceding lemma implies that
In order to complete the induction step we also have to cover the case i = 1, i.e., we have to prove (exp, (K))+ Y
+
IsrJr+l
.
( K )e x p , + , - , ( h ) 9
(10)
that is, we have to show that if we are given colorings
6 : C(expr(~))+l'+expr+ I
- i ( ~ )3
where 1 sisr + 1, then there is a set of cardinality K + that is homogeneous with respect to each f,. Obviously, there is a homogeneous set X of cardinality (expr(K))' with respect to the coloring f l . Then, using (9), we obtain that there is a set Y C X of cardinality K + that is homogeneous with respect to each f l , 2 5 i < r + 1. Then Y is homogeneous also with respect to f,. This verities (10). The proof is complete. 17. THE MAIN RESULTS IN CASE r = 2 AND K IS REGULAR; AND SOME COROLLARIES FOR r 2 3
The primary aim of this section is to study the partition relation K+ (v<);
in case K is regular. We shall again use tree arguments to establish our results. As we shall see later, more sophisticated arguments involving the canonical partition tree alone, which was defined in the proof of Lemma 15.2, would be sufficient. Our aim in defining also a different kind of tree is to give a wider illustration of tree arguments. The main theorem we are going to prove is as follows (we recall that L l ( ~is) the logarithm operation defined in Section 7):
(r
THEOREM 17.1. Let K, A,p and T be cardinals, and v: < T ) ordinals. Suppose that is regular, K 2 a,3 I I IK , 2 Ip IL I ( ~ )p,< K and, moreover, v , 2 2 jor any t t r . If P+(V<);
K
(the assumptions here allow the case T = 0 for the sake of convenience).
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TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
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PROOF. We may assume that 12Ko,p+. In fact, if the above assumptions are satisfied with some I , then they are also satisfiedwithY=i-p+.No. Assumewearegivenacoloringf: [ti]’+l i s . We have to prove t hat there is either a homogeneous set ofcolor 0 that has order tyw (or cardinality) I or a homogeneous set of color 1 i5 for some 5 < 7 that has order type V~ i1. We distinguish two cases: a ) I < K and b) A= K. Ad ma). We use the canonical partition tree defined in the’preceding section, but we impose additional stipulations in its definition. Namely, in the preceding section, we took an arbitrary enumeration S,.,, of the equivalence classes under E~.. Here we have a better choice. We shall have dom(S)=
u
“(lit),
USK+
and we are going to define the partition tree (K, S,R,7) by transfinite recursion. Assume that S l ( a i 1 ) has already been defined for some a
and, moreover, for any and
5 < 1 4 T, set
S,..,={XES(g‘)\
{s(g’)):f({s(g‘),x))=t)7 Sk)=Sg:g(a)
(4 ) (51
*
This completes the definition of the partition tree (K, S, R, T ) . For any h E T we have s ( h ) = min ( S ( h ) ) (61 and R ( h )= { ~ ( h ) )i.e., , 171
iR(h)l=l.
Observe that for any h~ T , the sequence (s(hPa):ccsdom ( h ) ) is a strictly increasing sequence of ordinals less than K ; this easily follows from (6). Fix now h E 17; and note that for any cr
f ( M h P a ) , s(h P S ) ) ) = h ( a ) ; in fact, this follows from (4) and ( 5 ) with g = h holds. Put
s(h ~
P
( a 4 l)j since
f lE)S(h ~ f lS(h) P~(a + I))= S ( g ) N h= {tE dom ( h ) :h(<)> 0) .
(8)
MAIN RESULTS I N CASE r = 2 AND K IS REGULAR; COROLLARIES FOR
r2 3
95
We may assume here that IN,l
In fact, the contrary easily implies that (2), which we want to prove. To see this, assume that (9) fails, and define the coloring E: [ N h ] ' - + z as follows: for any ct E N h write &(or) = 5
if and only if h(or) = 1 45 .
Then (1) entails that there is an ordinal 5 < t and a set X E N, of order type v: such that X is a homogeneous set of color 5 with respect to the coloring h'. The set Y={s(h " a ) : a ~ X u { d o m ( h ) ) f
has order type v: -k 1, since, as we pointed out above, after formula (7), the sequence ( s ( h PLY):LY
X=I\N,=
{< < I : h ( t ) = O f .
X has order type I in view of (3) and (9), and
Y = {s(g
5): 5 E xu { 1;)
is a homogeneous set of color 0 with respect to f according to (8). As Y has order type d 41, his completes the proof in casea), i.e., when R
1, (
V C1 ~
)~~~)'
(10)
in case I < K. Ad b). We have I = K in this case. We proceed similarly as in the proof of Theorem 11.3. Assume that there is no homogeneous set of cardinality K and of color 0 with respect to the coloring f : [K]'-~$'L. For every set X C K let H ( X ) be a maximal homogeneous subset of color 0 of X; the maximality of H ( X ) means that vx E X\H(X)3Y E WX)Cf({XYZ)#OI 9
(11)
96
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
and the assumption made just before implies
I HW)I< K .
(12)
We are going to define a partition tree (K, S, R, T). To this end, given S ( ( a + 1 ) f o r s o m e a < 9 = ~ + , whavetodefineS(g)foranygE(“+’)pprovided e S(gpa)#O (p will be specified later). Assume we are given such a g . Write g’=g pa, and put fib?’) = H(Sk’)) 9
and, noting that
R(g’)is a set of ordinals, write R(g’)=S(g‘)n (sup fi(g’)/ 1 ) ;
(13)
the point in choosing R(g’)in this way is that all elements of S(g’)\ R(g’) will then exceed any element of fi(g‘).Note that
I R(g‘)l< K
(14)
+
holds by the regularity of K in view of (12). For any x E fi(g’)and any color 1 5 with t < r , put \ R(g’):f ( { x y ) . ) =1 i5). . (15) sg,,K.x+s ={Y E To unravel the meaning of this definition, one should notice that x is an ordinal, and, as 5 < T < K (T < p [ < K] follows from (l)), the second subscript K . x 4 5 unambiguously identifies the ordinals x and 5 (the multiplication here means ordinal multiplication). If 9 is an ordinal which cannot be represented in form K . x i t for any x E fi(g‘) and t < T, then put S,.,,= 0. Writing p = K * K (ordinal multiplication again), it is clear that S,,,=O for any q 1 p . ( 1 1 ) implies that
which means that the definitions, given in (13)and (16),respectively, of R(g’) and S(g), are compatible (cf. (14.3)).Noting that W(g’)= H(S(g‘))has cardinality less than K in view of (12),we can see that there are fewer than K ordinals of the form K . x i 5 with x E A(g’)and 5 < T ; that is, there are fewer than K nonempty ones among the sets SB.,,, i.e., I ims (g’)!
MAIN RESULTS IN CASE
r = 2 AND K
I S REGULAR; COROLLARIES FOR
the dot. If S ( g ) # O , then define s ( ~ ) EB ( g " a ) and equation d a ) = K . 4 g ) i 3k)
,f(g)<7
r2 3
91
with the aid of the (18)
9
where the multiplication here again means ordinal multiplication. Note that we have dom ( g ) = a i 1 here; the functions s and fwill be defined only for those h E T for which dom (h) is a successor ordinal. The definitions of the partition tree ( K , S, R, T) and of the auxiliary functions s and f a r e complete. Apply Theorem 14.3.a with 1 4p = p + 1 replacing 1.The assumptions of this theorem are satisfied in view of (14) and (17), and in view of the fact that we have K @ < K for any K~ < K and p o < p 1; this latter assertion means the same thing as p 1 5 L,(K),which holds by our assumption p s LA(^). In fact, we have A = K in the present case, and L,(K) is infinite according to Theorem 7.2. So Theorem 14.3.a implies that T has a path of length 1 4 p i 1. Choose a h E T with dom(h)=l i p . Write x,=s(h p ( a i 1 ) ) i f a
+
+
<
for any a < p, where fwas defined in (1 8). It follows from (1 ) that there is a < 7 and a set Y C X of order type v z such that Y is a homogeneous set of color 5 withrespect to&. (15)and (18)imply that Y u (x,,; isa homogeneousset ofcolor 1 4 with respect to the coloring ,f. This set has order type v:+ 1, which completes the proof. We now turn to discussing the corollaries of the above result.
<
COROLLARY 17.2. Let K, A, p , u, and z be cardinals, and assume ASK, p , T < L ~ ( K ) < and K , a
( L A ( K ) L (P)JZ*
K
~
isWregular, (19)
PROOF. The result follows from the above theorem with the aid of the simple
relation
LA(Kk+((LA(K))m
WA1
(it is harmless to assume p 1 2 here so as t o satisfy the requirements of the above theorem). COROLLARY 17.3. Assume
7 Combinatonal
K
i s inaccessible and p , 7 < K . Then
98
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
PROOF. According to Theorem 17.1, one has to observe only that (P.T)++tp)f.
Note that in the present case we have L,(K)=K, and this is the case that was not covered by the preceding corollary.
COROLLARY 17.4. Assume A 2 2 , p >_ o,and T < cf ( p ) . Then @P)+
(cf ( P ) ) J 2 ’
+((JP)+,
(21)
PROOF. Write K’=A‘= (nu)’, d = ~and , p’, .r’=O. Apply Corollary 17.2 with the parameters having primes. The assumptions there are satisfied. In fact, L,.(K‘)is regular in view of Theorem 7.15 and so, using Theorem 7.17b, we obtain that o’=T
(22)
) )required , by the assumptions ofCorollary Since K’=J’, we have a’
1
(p’)J2.
Noting that we have cf ( ~ ) I L , . ( K ’ ) = L ~ .according (K’) to (22), relation (21) follows. The proof is complete. As a special case, we have
COROLLARY 17.5 (Erdos-Rado [1956]). Assume 1.22 and p 2 0 . Then
Another interesting consequence of the above theorem is COROLLARY 17.6. Assume 1.2 3, p L o,z I < cf ( p ) , and cr,
T~
< p. Then
PROOF. Noting that PI L,((?.e)+) according to Theorem 7.17.a,we can apply Theorem 17.1with (i.P)+, i.P, and p replacing K , 2, and p ; thedesired result follows from the simple relation P-+((P)?,
9
(4JZ.
COROLLARY 17.7. Assume 1. is a strong limit cardinal and p, T <1,. Then j.+(p)f.
(25)
MAIN RESULTS IN CASE r = 2 A N D K IS REGULAR: COROLLARIES FOR
r23
99
P R O O F . The case i.=w is covered by Ramsey's theorem (Theorem 10.2); therefore assume R>o.Let a = p . z . w ; then
(2")+ (PI5 -+
in view of (23). The desired result follows from here, since we have 2"
COROLLARY 17.8. Assume GCH and let A T w ,
T
and PROOF. (26) follows from (21 ) with p = E., and (27)follows from (24)with p = E., = t " l , and o,=r,=O. Using the Stepping-up Lemma (Lemma 16.1), we are now in a position to derive many interesting results from the above corollaries in the case r 2 3 . For the convenience of the reader, we recall that the operation exp is defined recursively by putting
T,
expo (O)= 2 and for any cardinal
;i and
exp, + ( i )= exp"(2')
any integer n.
GROLLARY 17.9. Assume 3.22, p>w, ~ < c( pf ) , and
let
r z 2 he an integer. Then
PROOF. For r = 2 these resultsareidentical to (21)and (23),respectively,and for r > 2 they follow by induction, with the aid of the Stepping-up Lemma. By applying the Stepping-up Lemma to (24), we get
COROLLARY 17.10. Let A, p, o, and T~ be cardinals satisfying i.23, p l w , r , < cf ( p ) , and (T, T , < p , and let r 2 2 be an integer. Then
COROLLARY 17.1 1 . Let p 2 w he a cardinal and r I2 an integer. Then
7*
100
TREES AND POSITIM ORDINARY PARTITION RELATIONS CH.
Iv
and so, a fortiori, (exp,- 1( P ) )+
+
fP + 5 .
(32)
Note that (32) can be directly obtained from the trivial relation P+
+
(P+ );
by induction, using the Stepping-up Lemma. (31) is, however, much stronger, showing that Theorem 17.1 has a special feature not contained in the Steppingu p Lemma. The positive results are in a sense weaker for r > 2 than for r = 2, since all results of the form K+(. . . )' with r 2 3 and K reguiar that we can prove are obtained by repeated applications of the Stepping-up Lemma starting with the case r = 2, except if K has some large cardinal property (for this latter remark cf. e.g. Theorem 29.5). 18. A DIRECT CONSTRUCTION OF THE CANONICAL PARTITION TREE Let K and t be cardinals, fix acoloring f: [ K ] < O + t , and consider the canonical partition tree (K, S, R, 7')associated with f; see the beginning of the proof of Lemma 15.2 for the definition. Simultaneously with this partition tree we also defined a function s: T + K . One can show that s is 1-1and onto and, moreover, that for anyg, h E T w i t h g c h we have s ( g ) s s ( h ) .(We in effect showed the latter assertion in the proof of Lemma 15.2;we omit the simple proof of the former here since we need these facts only in order to give some background to what follows.) So we can define an ordering <, of K which turns K into a tree isomorphic to (7: c ) by putting
5
(1)
and here 5 < / q implies g < q . It is obvious that the tree (K, <,) is just as applicable in proving the existence of large homogeneous sets as is the partition tree ( K, S, R, T). In fact, we shall see that there is some advantage in taking a closer look at the tree (K, if): by so doing we shall be able to prove (a strengthening of) Theorem 17.1 in case b; in the proof we gave in the preceding section we had t o define another partition tree. An important feature in considering the tree (K, <,) is that there is a simple direct definition of the partial ordering
101
A DIRECT CONSTRUCTTON OF THE CANONICAL PARTITION TREE
DEFINITION fg.1. cj Let X G K. Define the equivalence relation a--hxP
-l;
by putting
iff V u ~ [ X ] ~ ~ f ( u u ~ a ~ ) = f ( u u ~ ~( 21) )
for any a, p < K. (ii) For each a < K we define the function g,: a-2 by transfinite recursion on 5
(iii) For any a, P < K put For any a < K, introduce the notation
X u = { p : /3<,a)
= {p
g,(/?)= 1).
Note that we have
Xu.<=X , n 5 ,
(7)
where Xa,
<, so defined. (ii) ( K , <,) is a tree.
LEMMA18.2. (i) I f a < @ and gs(a)= 1, then gsPa=g,. (iii) For any a, B < K and u with B < p and u E [ X , A / ~ ]
(8)
that is, for any u E [ X , u {a)]
PROOF. Ad (i). Assume that gpf t=g,rt holds for some <
gs(t)=g,(5)
(10)
also holds, and the desired result will follow by transfinite induction. We have a --hxp.. according to the assumption gs(a)= 1; so we have a fortiori a =-hxp.
9
since X t , c X#,,. ~ Noting that X#,<= Xu,< in view of (9), this implies that t =hXp,t fl holds just in case 5 =xxm,t a holds; in other words, g # ( ( )= 1 just in case gu(t)=1, which proves (10).
I02
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
Ad (ii). First we prove that <, is a partial ordering, i.e., that it is an irreflexive, antisymmetric, and transitive relation. O . We are going to prove a stronger result: the assumptions of Theorem 17.1 with ; I = K > o imply the following: given an arbitrary coloring f : [ ~ ] ' - 1 $T, there is either a homogeneous set of color 0 that is stationary in K, or there is a 5 < T such that there is a homogeneous set of order type v: 4 1 and of color 1 i5. Symbolically, this assertion might be expressed as K+(Stat
(K), (Vc$
(11)
where Stat ( K ) denotes the set of stationary subsets of K. Here is, however, a note of caution: the notational conventions introduced in Section 8 are inadequate for explaining the meaning of this relation. For the proof, let f: [ K ] ~ - +1 i r be a coloring, and consider the tree ( K , <,>. (Strictly speaking, (K, <,) has not been defined, since dom ( f ) is a proper subset of[^]'". So put f'(u)=f(u)whenever u ~ d o m (f), and put f'(u)=Ootherwise, i.e., when u E [K)
holds for any a < K. In fact, assume on the contrary that 1 Y, 12p for some a < K . Then (17.1) implies that there is a 5 < z and a set Z E of order type vt such that f ( { y a } ) = l i ( holds for any ~ E ZIt. follows from (8) that Z u { a ) is a
A DIRECT CONSTRUCTION OF THE CANONICAL PARTlTlON TREE
103
homogeneous set of color 1 45 with respect to f . As Z u {a; has order type v,; 1, this completes the proof of the theorem in case (13) fails for some a < K. Assume therefore that (13) holds for all a < K. Put o = w if p < w, o = p is p is regular, and o = p + if p is singular. Note that o is a regular cardinal and o < K. In fact, since we assumed K > w and p < K, the case o = IC could occur only if p was singular and p + = K; but then L , ( K )Icf (p)< p, which conflicts with our assumption P < L ~ ( K ) = L , ( K( L) = K in the case considered). Put A = {a< K : cf (a)= 0 3 . A is a stationary subset of K as it contains the oth element of an arbitrary club in K. The function
h(a)= sup Y,
is a regressive function on A in view of (13). Fodor’s theorem (Theorem 5.4) implies that there is an ordinal C < K and a set B s A stationary in K such that h(a)=< holds for any a € B. Writing f,(/?)=f({a@) for any a, fl wit?/?
S, r Y,=L in particular
K = Y, for any a E C. We claim that C is a chain of the tree (K, <,).
(14)
In fact, let a, /3 E C with a
X,ny=X,ny
f({daj)= f,(4=
fm=f ( { S P ) - ) ;
the second equality here follows from (14) and (12), since if 6 E Y. = Y,, then &(a) and f,(6) coincide, and if &#Ye= Y, then f , ( S ) = f , ( 6 ) = 0 . Using (8), we obtain that f({6y))=f({6a).) since 6
104
TREES A N 6 POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
We claim that C is a homogeneous set of color 0. In fact, if /? E C then Ply,= y
(equality holds here by (14)), since Y,c /? (cf. (12)).Let now a, /?E C be such that P
B € xu\Y = xu\y.. Hencethedefinitionof ruin (12)impliesf({a/?))=O;thisshowsthat Cisin fact a homogeneous set of color 0. As C is stationary, this completes the proof of the theorem. The reader may easily see that we even proved slightly more than (11). We illustrate the situation only in a special case when K = I = K2,p = K,, T = K1. The above proof gives e.g. the following: Assume 2N0= K,;let D be a stationary set in K 2 with Ds{a
(15)
and let f: [ D I 2 - 2 be a coloring. Then there is either a set C , G D stationary in K 2 with f " C O = (0) or a set C , of cardinality K, with f"C1 = { 13. Using ideas of J. E. Baumgartner [19763, one can show that this assertion is false if we modify our assumptions by requiring that instead of (15).
Dc{a
TREES AND POSITIVE ORDINARY PARTITION RELATIONS
A general framework for tree arguments is presented, and this is applied in order to derive the main results of the form K + ( & ) : < s , where K is a regular cardinal upon which no “large cardinal assumptions” (see Chapter VII) are imposed. 13. TREES
We defined trees in Section 10, where we needed the concept for the second proof of Ramsey’s theorem, but, for the sake of completeness, we recall the definition here : A tree is a partial order (7; <) such that, for any x E 7; the set of its predecessors, pr (x)=pr (x, (T, <))={y E T : y < x ) is wellordered by < (we assume that < is irreflexive, it., that 7x < x holds for any x E 7;though this is not an important assumption). We sometimes write T instead of (7; <).For any x E T, the order o(x)= o(x, (IT; <)) of the element x is the order type of pr (x). For any ordinal a, the set T[a] = (x E T: o(x)= a ) is called the ath level of T, and Tla={x~ T: o(x)
81
TREES
THEOREM 13.1. Let (7: <) be a tree, and assume that K I IT1 is regular. Let I be a cardinal with w s i. < K . Then T has a path of length 2. 4 1 provided one of the fbllowing two conditions is satisfied. a) I Is @)I < K holdsjor every path p of T, and < K,for every K, < K and i., < j.. b) There is a cardinal CT with a?< iisuch that I Is ( p ) (5 d[P1""1 holdsfor eoery path p of T. (Jf A>w, rhen one cun write CTIPI" instead Of'&"""1 here.)
PROOF. We have to prove that T has length 2 j. 4 1, as this is clearly equivalent to the existence of a path of length j.4 1; i.e., we have to show that TI;.# T As T was assumed to have cardinality 2ii, it is enough to show to this end that ITILI
As I < K and K is regular, it will be enough to show that 1 TI a [< K holds for any
a
contradicts our choice of q, establishing b). (The remark in parentheses can be proved similarly; or, alternatively, it is a direct consequence of the main result if we add o new levels at the bottom of T.) The proof is complete. Another result establishing the existence of long branches is THEOREM 13.2. Let K, I , p, and t be cardinals, and suppose that K is regular, (0< )T < p < I , and d p < K. Let 9 be an ordinal, and let T be a set offunctions f E "r, and assume that if f e T , then f " a E T for any a s 3. T ordered by
0I A,
u
as9
6 Combinatonal
82
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
inclusion is a tree. Assume that 1 TI 2 ti and that
I (tE dom ( f )f(O>o)l : holds for
any f~ 7: Then T has
Q
path
(1)
of length 1 i1.
PROOF. It is obvious that T is a tree. As before, it will again be sufficient to prove that l’Tlj.]
The number of functions f
E
U ‘7
satisfying ( 1 ) is I
a
i.e., l T l q [ < ~which , we wanted to prove.
14. TREE ARGUMENTS
One is often interested in constructing a ‘long’ sequence ( x a :a < 8 ) ofelements of a set S that has ‘nice’ properties. For example, in the proof of Ramsey’s theorem we were given a coloring f :[S]‘+k, and we constructed a sequence ( x i : i < w ) in such a way that we had f ( u u { x , ) ) = f ( ~ u ( x , whenever ~) m l n < w and u ~ [ ( x ~ : i < m ] ] ‘The - ~ . procedure we followed In the second proof of Ramsey’s theorem was that, when we had already constructed a finite sequence { x , : i < l ) that had the above property (with m s n < l ) , then we continued this sequence in several ways that, in a sense, represented all possible continuations, thus ensuring that at least one of the sequences had an infinite continuation. In constructing these sequences, it was helpful to define the set S((x, : i < I ) ) of the potential successors of the sequence ( x i : i < 1). The situation in Ramsey’s theorem was simplified by the fact that we did not have to consider sequences of transfinite length. The proof of Theorem 11.3 above was in many ways analogous, though an additional complication was caused by the fact that the elements of the ‘long sequence’ to be constructed were subsets, rather than elements, of the underlying set. We are now going to consider a general framework for arguments of this type, called tree arguments, and we shall prove a few results concerning the sizes of ‘nice’ sequences that can be obtained in this way.
83
TREE ARGUMENTS
The underlying concept of tree arguments is that of a partition tree, also called ram$fication system: DEFlNi-rioN 14.1. A partition tree is a pair ( E , S ) , where E is a set (called the underlying set), and S is a function with the following properties:
(i) ra ( S ) s Y ( E ) ;(ii) dom ( S ) =
u<>
a
' p for some ordinals 9 and p; (iii) S(O)=E
(notethatOE'p= ( O ) ) ; ( i v ) i f J ; g ~ d o(mS ) a n d j s g , t h e n S ( g ) s S ( f ) ; a n d(v)if f ~dom (S), and q=dom ( f )is a limit ordinal, then S ( f ) =
n S(,f ' a ) .
a
S will usually be defined by transfinite recursion with the aid of the recursion for mu la
for any ordinal 5. Here G is a given function whose range is a subset of Y( E ) and whose domain is what it ought to be according to (1). In order for (1) to define a partition tree, G has to fulfil the following conditions with X = Sldom ( f ) : G(O,O)= E ,
(iii')
if q =dom (f )is a limit ordinal, so that (iii), (iv), and (v) of the above definition be satisfied. If these conditions are satisfied, then (1) obviously defines a partition tree. (iii') and (v') mean that we have no freedom at all in specifying G ( f ,X )unless dom ( f ) is an ordinal of form a $ 1 for some a; that is, in order to define a partition tree, we have to specify G(f; X ) only in case Jom ( f ) is a successor ordinal. We are going to define some auxiliary concepts concerning a partition tree ( E , S). Noting that c always denotes strict inclusion here, for any f ~ d o m( S ) write (3 1 N f ) =S(f)\ {sk):fCgE dom 6))
u
and, moreover, put
T = { f E d o m ( S ) :S ( f ) # O )
In view of (iv) of the above definition, Tordered by inclusion c is a tree (this 6'
84
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
justifies the word tree in the term partition tree), and we have R(J')=S(,j')\U(S(g):,f~g~T&dom(g)=dom(f')/l).
Our main interest below will be how long the branches of T are, as a branch of T will define a 'nice' sequence of the same length. We shall occasionally refer to a partition tree as a quadruple (E, S, R, T), where R and T are defined from S in the way described above. We now give some explanation about the intuitive meanings of the letters E, S, R, and 7: In the simplest case, we want to construct a long 'nice' sequence (xu: a < q) of pairwise distinct elements of E. We construct this sequence step by step, making alternative choices at each step, the number of alternatives being Limited by IpI, where p is the ordinal in (ii) of the above definition. Having constructed a sequence ( x ~ c(: < 5 ) of length t, a function .fi 5-p keeps track of the alternative choices made, and the sequence (xg:a < t )is obtained in the form ( s ( f " a ) :a < ( ) , where s is a function from T into E defined by transfinite recursion (simultaneously, or subordinated to, the recursive definition of S as T one described in (1)) in such a way that s ( f ) E S ( , f ) holds for any , ~ E Usually, can pick an arbitrary element of S ( f ) as s ( f ) , that is, S ( , f ) is the set of potential successors of the sequence ( s ( f " a ) : u
LEMMA 14.2. For any partition tree ( E , S, R, T ) we have E = ( j { R ( f ) :f e T : .
85
T R t E AR(illMF.NTS
PROOF. For an arbitrary x E E put T , = { y E T:xES(y)}.
This set is not empty in view of (iii) in Definition 14.1, and (T,, c ) is obviously a tree by (iv) there. By Hausdorffs Maximal Chain Theorem (see Subsection 4.7), there is a branch b of this tree; put f = b. We claim that , f ' ~h. This is obvious if b has a last element in the ordering c ,as this last element can only be ,f.Hence, assuming that ,f$ h, 5 = tp ( h , c ) = dom (.f ) must be a limit ordinal (we cannot have <=O, since h is not empty, as Tv #O). Then
u
xE
( q g ) :g E b; =
n (s(,jh a ) :a < < ) =s(f'),
where the last equality holds in view of (v) in Definition 14.1; observe that (ii) there is also used to show , f dom ~ (S)(this is why it is an important technicality ,'u ). So . f ~T,; therefore
to take dom (S)= (J "p instead of dom (S)= US 9
f$ b
U
means that b u {f). is a chain of T, properly including b which contradicts the assumption that b is a branch. This establishes the claim f E b c T,. Hence x E S(f ). I f g is an element of dom (S)that properly includes f then x $ S k ) , since otherwise b u {g)would be achain of T, that properly includes b which would be a contradiction. Hence x E R ( , f )holds by the definition of R in (3)above. As x E E was arbitrary, this completes the proof. The main theorem of this section is simply a reformulation of the results of the preceding section in the light of the lemma just proved. We recall that ims (S, T) denotes the set of immediate successors in T of the element f: In the present case we have ims (,A T ) = { g E T :fcg eL dom (g) =dom (f) i 1 3. THEOREM 14.3. Let ( E , S , R, T ) be a partition tree; let K be a regular cardinal with K 5 1 E I, and assume that IR (f) 1
u
asY \
PROOF. I R ( f ) l < K implies I TI 2 K by virtue of the preceding lemma. Hence the result is a simple restatement of Theorems 13.1 and 13.2. Note that each path of limit length has at most one successor in case of a partition tree (E, S, R, T ) .
86
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
Hence, in cases a ) and b), the inequalities for 11s ( p ) l required in Theorem 13.1 follow from the analogous inequalities for lims ($)I here. There is a possibility of generalizing the concept of partition trees to cases when S (,f) assumes values in any partially ordered set instead of the power set of a given set E . A case in point is e.g. the following simple result observed by K. Kunen: given a cardinal K, any (K', 3)-distributive complete Boolean algebra B that satisfies the K-chain condition is atomic. To prove this result, one has t o use partition trees where S ( f ) E B. (For the definitions of the concepts here, see e.g. Sikorski [1960]; (K', 3)-distributivity here is t o be understood in the sense of ( < K', < 3)-distributivity, and not of I( K', I 3)-distributivity.)
15. END-HOMOGENEOUS SETS Let a be an ordinal and T a cardinal, and let ,fi [a] <"-+T be a coloring. Call a set X c a end-homogeneous with respect to ,f iffor every finite set u c X and for every p, v E X with max u < p, Y. we have f ( U U
(p))=.f(uu
{\I]).
(11
I f f is only a partial coloring of [u] < w i.e., iff is a function from a subset of [a] < w , then call X E end-homogeneous ~ with respect t o .f if ( 1 ) holds whenever u E [XI <", max u < p , v , and u u { p i , u u E dom ( , f ) . The partition relation a+
(8)r
(2)
means the following: for every coloring ,$ [a]""+r there is an endhomogeneous set of order type p. End-homogeneous sets are very important in establishing positive partition relations; in fact, they were already used in both proofs of Ramsey's theorem in Section 10. The following simple lemma throws some light upon the situation. LEMMA 15.1. Let r 2 1 be un integer, 7 u cardinul, ond let a, b, 1,: (( < T )be ordinuls. Assume p+(\a:);:;. (3 1 Then (r -+(p), implies
and a+ ( p 4 1), implies
(r-+(\j:):.
u+(\,:+
1
I,;.<,.
(4) (5)
An instance of relation (4) was used above in the proof of Ramsey's theorem, and relation (5) will be uskd below in the proof of the Stepping-up Lemma (Lemma 16.1).
87
END-HOMOGENEOUS SETS
PROOF. Let ,/’:[ c r ] * - t ~ be a coloring. First we verify (5). Assume to this end that cr+(fiit), holds; then there is a set X of order type p i 1 that is end-homogeneous with respect to .f: Let i be the maximal element of X, write X ’ = X \ !(),and define the coloring g: [X’]*-’+T as follows: put g ( u ) = = , f ( u u (0 )for every u E [X’]‘- ’. By ( 1 ), there is a < T and a set Y E X’of order type 11,. such that Y is homogeneous of color 5 with respect tog. i.e., g”[ Y ] ‘ - ’ = = (0.But then we have
<
, f ( i ? ) = , f ( ( r \ (max
vI)u(0)
for any I’ E [ Y u (<;]‘by the end-homogeneity of X, and the right-hand side here equalsg(z-\ (max I . ) ) = by < the definition ofg. Hence Y u (i) is a homogeneous set of order type v , . i 1 and of color with respect to ,I; which proves (5). In the proof of (4), assume first that fl is a limit ordinal. The relation ~ c - - + ( f i ) ~ implies the existence of an end-homogeneous set X 5 CI of order type with respect t o the coloring ,J [cr]‘-*r. Given an arbitrary u E [XI‘-’, put g ( u ) = = , f ’ ( u u ( v i ) for a V E Xwith v>max ( u ) ; there is such a 11, since t p X = p was assumed t o be a limit ordinal, and g ( u )does not depend on the choice of vin view ofthe end-homogeneity of X. By (3). there is a 5 < T and a set Y G X of order type Y: that is homogeneous of color 5 with respect t o the coloring g : [XIr-’+?: clearly, Y is then also homogeneous ofcolor with respect to .I:This establishes (4) in case p is a limit ordinal. If B=p i1 for some /?’,then we have t o work with the last element of X as we did in the proof of (5). It is, however, simpler t o observe that (3) implies
<
<
1,r;f
P’+(v,
in this case (indeed, by taking away the last element of p, we can take away only the last element of a homogeneous set), and so we have cr+
( ( I #
1) i 1 );
<
~
in view of ( 5 ) with p’ replacing p. The proof is complete. The following lemma, due t o ErdBs and Rado, confirms the existence of large end-homogeneous sets. It will usually be applied together with the preceding lemma t o prove the existence of large homogeneous sets. I t is, however, also important in itself: in Section 45 we shall give some applications of it concerning set mappings in cases where the corresponding partition relations would lead to weaker results. LEMMA 15.2. For any cardinals
K 2w
and i L 2 Mie hare
(E.F)+--t(K/
l)i.
(6)
88
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
PROOF. Writing E = (AF)+, let f : [ E l <"--+A be a coloring. We are going to define a partition tree ( E , S, R, T ) , called the canonical parfitionfreeassociated with the coloring .f: To this end, write dom(S)=
u
(7 1
'p,
US9
where p and 9 will be specified later. As we mentioned in the preceding section, we can define S by the recursion formula
k)),
S(g)=G(g,
where G has to be specified only in case dom (g) is a successor ordinal and S(g * 5 ) # 0 for any 5
5 E S(g')\
(9)
/?
PP):
{~(g')',, define the coloring
,fg,.t: [E,.]<"-A
&.& 1=f (u u { t 1) for any u E [E,.]
<"'.
For any two
by putting (10)
5, q E S(g')\ {~(g')),put
E ~is .obviously an equivalence relation. The number of equivalence classes under is I the number of possibilities for the coloring &.,r, i.e., is at most =g6
where we recall that a = d o m (g'). Let S,.,,, v < p , be an enumeration of the Here . p (cf. (7)) has to be large enough so that we equivalence classes under E ~ . may enumerate all the equivalence classes (as these classes are pairwise disjoint, p = E = I El clearly suffices); if I p Jis larger than the number of these equivalence classes, then put S,,,=O as many times as necessary. Write
S(d =&@ ,)
f
and, for further reference, if S(g)#O, then write fg
=f,,.<
9
END-HOMOGENEOUS SETS
89
where 5 E S k ) ; clearly, f, does not depend on the particular choice of 5 here. By (lo), we have fe(u)=f(uu
{tl)
(14)
for any finite set u s {s(g * P ) : j < d o m (g)) and any ~ E S C ~This ) . finishes the definition of S. Choose 3 so large that S ( g ) = O for any g ~ ' p clearly, ; 9= E + suffices since, for any g with S ( g ) # O , the set {s(g p a ) : a s d o m (g))
consists of pairwise distinct elements of the cardinal E. Putting, as usual, T = { h Edom (S): S(h)#O). , we clearly have R ( h ) = { s ( h ) )for any h E 7; i.e., we have IR(h)l= 1
(15)
for any h E 7: Taking h =g' in (12), we can conclude that lims ( h ) l s AI[lhl+'l''"l
(16)
holds for any h E 7;where ims (h) denotes the set of immediate successors of h in T; in fact, exactly one immediate successor of h in T corresponds to each (nonempty) equivalence class under -,,. Noting that 1 El = E = (,IF)+, we can conclude from Theorem 14.3.b with (A!)', K, and 1replacing K , I, and a, respectively, that T has a path of length K + 1, i.e., that there is a g E T with dom (g)= K. Write 5,=s(g . first that for each M ~ KObserve (8), we obtain that
pa)
t,
5, = min S(g + a )< tP provided a <
K,
since 5pE
Sk * P I G S k " a ) \ {t,)
holds by our construction above. Hence the set X = { 5,: 1Ia s K ) . has order type K / 1. We claim that X is an end-homogeneous set with respect to f. In fact, if u E [XI and max u = 5, < t p tY , (or a= 0 and t P , E X in case u is empty), then we have a 4 1s P, 7, and so to,tYE S(g (a 41)). Hence (14) implies that
ry
f ( u u { 5 P ~ ) = f , ~ ( , ~ l , ( u ) = f (it),;) uu holds; this shows that X is indeed end-homogeneous. The proof is complete.
90
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
COROLLARY 15.3. Let
K
and z be cardinals with z < K and
Iv
K > (11.
Then
iI),
(2")++ ( K + and
I),
(2")++
hold.
PROOF. For (17), replace E. and K with 2" and K + , and, for (18), with z and K , respectively. (18) will be used for the proof of the Stepping-up Lemmas in the next section; (19) will be used in connection with set mappings (see Section 45), and not in order to establish a partition relation. 16. THE STEPPING-UP LEMMA
The following lemma, one of our most important tools in establishing positive ordinary partition relations, is an easy consequence of the results of the preceding section:
LEMMA16.1 (Stepping-up Lemma). Let integer, and v: ( 5
~
and W
5
be cardinals, r 1 2 an
(V:);f
ti+
holds, then we haue
K
(2"++(vci l)?<,
PROOF. We may assume that v: 2 r holds for all t < z (cf. Subsection 9.5). In this case, noting that K is infinite and r r 2 , (1) implies that T < K ; hence, we have (2"+
+(ti
i1 ),
according to (15.18) in Corollary 15.3. Hence (2) follows from (15.5) in Lemma 15.1 with (2!))+ and K replacing 01 and 8, respectively. The proof is complete. This result alone has nontrivial corollaries. As an illustration, observe that the relation
(Po)+
+
(K,I;,
which is a theorem of Erdos and Rado, follows from the trivial relation Nl+(N,)A. This is, however, not the appropriate place to go into discussing the corollaries of the above lemma, since as we shall soon see, stronger results can be obtained in the case r = 2 by more refined methods. This will be done in the next section.
91
THE S T E P P I N G U P L E M M A
The aim of the rest of this section is t o derive a result saying that certain partition relations hold simultaneously; this result will have an importan1 application in the proof of the General Canonization Lemma, given in Section 28. We start with the following
DEFINITION 16.2. Let CI and p be ordinals, N a set of integers 2 1, and cardinals. Then the partition symbol
'si
~ (Ei N
11
called the simultaneous ordinury purtition symbol and read as " a arrows simultaneously for i E N with.ri colors", is said to hold if we have the following: Given a n arbitrary coloring ,h: [ c x ] ~ + T ; for each i E N , there is a set X c CI of order type p that is homogeneous with respect to each ,fi, i E N , i.e., for any i E N there is a { < r , with .f';'[Xli= it;.T h e negation of the above symbol will, as usual, be written as ay(p):;" . We mention a few natural variations of the above symbol. Where k and 1 are integers, the relatidn
CIy (/jf,< means the same thing as
as2(pi; I j
I
<1
~ S I < I ~
The meaning of the symbol
$3( pf,,, . ..k" ., IA".
will be defined as
Note also that
.SI:m(fi):'
,
--.I!..
Ik". . . .k"
I;
ol"z(p):F"' means the same thing as a-(fi )Zw (cf. Subsection 8.6).
Using the ideas ofthe proof of Lemma 15.1, it is easy to establish the following:
16.3. Let r( 2 1) be an integer, LEMMA cardinals. Assume that
-(Pi holds with ~ = m a x{ ' s i : 2 < i < r ) . Then
CI,
b, and
I>,
v ordinals, and zi ( 2 5 ; i S r )
(3)
92
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
implies
Iv
aszlf+m(v+ 1 f,si
PROOF. For each i with 2 < i < r , let ,h:[ali+7, be a coloring. Put , f =
u ,f;;
Z i t r
then ,f maps a subset of [a] into 7 = maxZsicr7i.By (3), there is a set X E a of order type 4 1 that is end-homogeneous with respect to f: Write X = X'u{C}, where 5 is the maximal element of X. For each integer i with 2 < i 1 r define the coloringg,: [X']'-'+rbyputtingg,(u)=fi(uu (5))foranyu E [X']'-'.Then, by (4), thereis a set Y E X of order type v that is homogeneous with respect to each g i , 2 5 i s r . We can now conclude in exactly the same way as we did in the proof of (15.5) in Lemma 15.1that Y u { [). is a homogeneous set of order type v i1 with respect to each coloring ,h, 2 <_ i 5 r. The proof is complete. We can now prove the following:
LEMMA 16.4 (Simultaneous Stepping-up Lemma). Let r bean integer, t i ~ and w T, 2 1 (2 5 i 5 r ) cardinals, and v an ordinal. Assume that
holds. Then we have (2"+sz(,&
1);,-*
PROOF. We may assume r 2 2 and v 2 r here (in fact, if e.g. 2 1 v < r, then we can replacer with r ' = v in ( 7 ) ) . Then (6) implies that T ~ < Kfor each 7 with 2 1 i 1 r ; hence we have ( 2 " ) + + ( K / l), with ~ = m a x ~ ~according , ~ , . t ~ to (15.18). Using now the preceding lemma with a= (25))+ and p= K, we can therefore see that (6) indeed implies (7). The proof is complete. Define the operation exp by induction as follows: expo ( K ) = K
and
exp,,, (K)=exp,, ( 2 " ) ,
where K is a cardinal and n'runs over integers. The result we are aiming at (the one we shall need in the proof of the General Canonization Lemma in Section 2 7 ) is the following:
COROLLARY 16.5. Let r 2 1 be an integer and
K
~
aOcardinal. Then we have
MAIN RESULTS I N CASE r = 2 AND K IS REGULAR; COROLLARIES FOR
r23
93
PROOF. For r = 1 the assertion says K + - - + ( K + ) ~ ,which is obviously true. If (8) holds with some r.2 1, then the preceding lemma implies that
In order to complete the induction step we also have to cover the case i = 1, i.e., we have to prove (exp, (K))+ Y
+
IsrJr+l
.
( K )e x p , + , - , ( h ) 9
(10)
that is, we have to show that if we are given colorings
6 : C(expr(~))+l'+expr+ I
- i ( ~ )3
where 1 sisr + 1, then there is a set of cardinality K + that is homogeneous with respect to each f,. Obviously, there is a homogeneous set X of cardinality (expr(K))' with respect to the coloring f l . Then, using (9), we obtain that there is a set Y C X of cardinality K + that is homogeneous with respect to each f l , 2 5 i < r + 1. Then Y is homogeneous also with respect to f,. This verities (10). The proof is complete. 17. THE MAIN RESULTS IN CASE r = 2 AND K IS REGULAR; AND SOME COROLLARIES FOR r 2 3
The primary aim of this section is to study the partition relation K+ (v<);
in case K is regular. We shall again use tree arguments to establish our results. As we shall see later, more sophisticated arguments involving the canonical partition tree alone, which was defined in the proof of Lemma 15.2, would be sufficient. Our aim in defining also a different kind of tree is to give a wider illustration of tree arguments. The main theorem we are going to prove is as follows (we recall that L l ( ~is) the logarithm operation defined in Section 7):
(r
THEOREM 17.1. Let K, A,p and T be cardinals, and v: < T ) ordinals. Suppose that is regular, K 2 a,3 I I IK , 2 Ip IL I ( ~ )p,< K and, moreover, v , 2 2 jor any t t r . If P+(V<);
K
(the assumptions here allow the case T = 0 for the sake of convenience).
94
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
PROOF. We may assume that 12Ko,p+. In fact, if the above assumptions are satisfied with some I , then they are also satisfiedwithY=i-p+.No. Assumewearegivenacoloringf: [ti]’+l i s . We have to prove t hat there is either a homogeneous set ofcolor 0 that has order tyw (or cardinality) I or a homogeneous set of color 1 i5 for some 5 < 7 that has order type V~ i1. We distinguish two cases: a ) I < K and b) A= K. Ad ma). We use the canonical partition tree defined in the’preceding section, but we impose additional stipulations in its definition. Namely, in the preceding section, we took an arbitrary enumeration S,.,, of the equivalence classes under E~.. Here we have a better choice. We shall have dom(S)=
u
“(lit),
USK+
and we are going to define the partition tree (K, S,R,7) by transfinite recursion. Assume that S l ( a i 1 ) has already been defined for some a
and, moreover, for any and
5 < 1 4 T, set
S,..,={XES(g‘)\
{s(g’)):f({s(g‘),x))=t)7 Sk)=Sg:g(a)
(4 ) (51
*
This completes the definition of the partition tree (K, S, R, T ) . For any h E T we have s ( h ) = min ( S ( h ) ) (61 and R ( h )= { ~ ( h ) )i.e., , 171
iR(h)l=l.
Observe that for any h~ T , the sequence (s(hPa):ccsdom ( h ) ) is a strictly increasing sequence of ordinals less than K ; this easily follows from (6). Fix now h E 17; and note that for any cr
f ( M h P a ) , s(h P S ) ) ) = h ( a ) ; in fact, this follows from (4) and ( 5 ) with g = h holds. Put
s(h ~
P
( a 4 l)j since
f lE)S(h ~ f lS(h) P~(a + I))= S ( g ) N h= {tE dom ( h ) :h(<)> 0) .
(8)
MAIN RESULTS I N CASE r = 2 AND K IS REGULAR; COROLLARIES FOR
r2 3
95
We may assume here that IN,l
In fact, the contrary easily implies that (2), which we want to prove. To see this, assume that (9) fails, and define the coloring E: [ N h ] ' - + z as follows: for any ct E N h write &(or) = 5
if and only if h(or) = 1 45 .
Then (1) entails that there is an ordinal 5 < t and a set X E N, of order type v: such that X is a homogeneous set of color 5 with respect to the coloring h'. The set Y={s(h " a ) : a ~ X u { d o m ( h ) ) f
has order type v: -k 1, since, as we pointed out above, after formula (7), the sequence ( s ( h PLY):LY
X=I\N,=
{< < I : h ( t ) = O f .
X has order type I in view of (3) and (9), and
Y = {s(g
5): 5 E xu { 1;)
is a homogeneous set of color 0 with respect to f according to (8). As Y has order type d 41, his completes the proof in casea), i.e., when R
1, (
V C1 ~
)~~~)'
(10)
in case I < K. Ad b). We have I = K in this case. We proceed similarly as in the proof of Theorem 11.3. Assume that there is no homogeneous set of cardinality K and of color 0 with respect to the coloring f : [K]'-~$'L. For every set X C K let H ( X ) be a maximal homogeneous subset of color 0 of X; the maximality of H ( X ) means that vx E X\H(X)3Y E WX)Cf({XYZ)#OI 9
(11)
96
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
and the assumption made just before implies
I HW)I< K .
(12)
We are going to define a partition tree (K, S, R, T). To this end, given S ( ( a + 1 ) f o r s o m e a < 9 = ~ + , whavetodefineS(g)foranygE(“+’)pprovided e S(gpa)#O (p will be specified later). Assume we are given such a g . Write g’=g pa, and put fib?’) = H(Sk’)) 9
and, noting that
R(g’)is a set of ordinals, write R(g’)=S(g‘)n (sup fi(g’)/ 1 ) ;
(13)
the point in choosing R(g’)in this way is that all elements of S(g’)\ R(g’) will then exceed any element of fi(g‘).Note that
I R(g‘)l< K
(14)
+
holds by the regularity of K in view of (12). For any x E fi(g’)and any color 1 5 with t < r , put \ R(g’):f ( { x y ) . ) =1 i5). . (15) sg,,K.x+s ={Y E To unravel the meaning of this definition, one should notice that x is an ordinal, and, as 5 < T < K (T < p [ < K] follows from (l)), the second subscript K . x 4 5 unambiguously identifies the ordinals x and 5 (the multiplication here means ordinal multiplication). If 9 is an ordinal which cannot be represented in form K . x i t for any x E fi(g‘) and t < T, then put S,.,,= 0. Writing p = K * K (ordinal multiplication again), it is clear that S,,,=O for any q 1 p . ( 1 1 ) implies that
which means that the definitions, given in (13)and (16),respectively, of R(g’) and S(g), are compatible (cf. (14.3)).Noting that W(g’)= H(S(g‘))has cardinality less than K in view of (12),we can see that there are fewer than K ordinals of the form K . x i 5 with x E A(g’)and 5 < T ; that is, there are fewer than K nonempty ones among the sets SB.,,, i.e., I ims (g’)!
MAIN RESULTS IN CASE
r = 2 AND K
I S REGULAR; COROLLARIES FOR
the dot. If S ( g ) # O , then define s ( ~ ) EB ( g " a ) and equation d a ) = K . 4 g ) i 3k)
,f(g)<7
r2 3
91
with the aid of the (18)
9
where the multiplication here again means ordinal multiplication. Note that we have dom ( g ) = a i 1 here; the functions s and fwill be defined only for those h E T for which dom (h) is a successor ordinal. The definitions of the partition tree ( K , S, R, T) and of the auxiliary functions s and f a r e complete. Apply Theorem 14.3.a with 1 4p = p + 1 replacing 1.The assumptions of this theorem are satisfied in view of (14) and (17), and in view of the fact that we have K @ < K for any K~ < K and p o < p 1; this latter assertion means the same thing as p 1 5 L,(K),which holds by our assumption p s LA(^). In fact, we have A = K in the present case, and L,(K) is infinite according to Theorem 7.2. So Theorem 14.3.a implies that T has a path of length 1 4 p i 1. Choose a h E T with dom(h)=l i p . Write x,=s(h p ( a i 1 ) ) i f a
+
+
<
for any a < p, where fwas defined in (1 8). It follows from (1 ) that there is a < 7 and a set Y C X of order type v z such that Y is a homogeneous set of color 5 withrespect to&. (15)and (18)imply that Y u (x,,; isa homogeneousset ofcolor 1 4 with respect to the coloring ,f. This set has order type v:+ 1, which completes the proof. We now turn to discussing the corollaries of the above result.
<
COROLLARY 17.2. Let K, A, p , u, and z be cardinals, and assume ASK, p , T < L ~ ( K ) < and K , a
( L A ( K ) L (P)JZ*
K
~
isWregular, (19)
PROOF. The result follows from the above theorem with the aid of the simple
relation
LA(Kk+((LA(K))m
WA1
(it is harmless to assume p 1 2 here so as t o satisfy the requirements of the above theorem). COROLLARY 17.3. Assume
7 Combinatonal
K
i s inaccessible and p , 7 < K . Then
98
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
PROOF. According to Theorem 17.1, one has to observe only that (P.T)++tp)f.
Note that in the present case we have L,(K)=K, and this is the case that was not covered by the preceding corollary.
COROLLARY 17.4. Assume A 2 2 , p >_ o,and T < cf ( p ) . Then @P)+
(cf ( P ) ) J 2 ’
+((JP)+,
(21)
PROOF. Write K’=A‘= (nu)’, d = ~and , p’, .r’=O. Apply Corollary 17.2 with the parameters having primes. The assumptions there are satisfied. In fact, L,.(K‘)is regular in view of Theorem 7.15 and so, using Theorem 7.17b, we obtain that o’=T
(22)
) )required , by the assumptions ofCorollary Since K’=J’, we have a’
1
(p’)J2.
Noting that we have cf ( ~ ) I L , . ( K ’ ) = L ~ .according (K’) to (22), relation (21) follows. The proof is complete. As a special case, we have
COROLLARY 17.5 (Erdos-Rado [1956]). Assume 1.22 and p 2 0 . Then
Another interesting consequence of the above theorem is COROLLARY 17.6. Assume 1.2 3, p L o,z I < cf ( p ) , and cr,
T~
< p. Then
PROOF. Noting that PI L,((?.e)+) according to Theorem 7.17.a,we can apply Theorem 17.1with (i.P)+, i.P, and p replacing K , 2, and p ; thedesired result follows from the simple relation P-+((P)?,
9
(4JZ.
COROLLARY 17.7. Assume 1. is a strong limit cardinal and p, T <1,. Then j.+(p)f.
(25)
MAIN RESULTS IN CASE r = 2 A N D K IS REGULAR: COROLLARIES FOR
r23
99
P R O O F . The case i.=w is covered by Ramsey's theorem (Theorem 10.2); therefore assume R>o.Let a = p . z . w ; then
(2")+ (PI5 -+
in view of (23). The desired result follows from here, since we have 2"
COROLLARY 17.8. Assume GCH and let A T w ,
T
and PROOF. (26) follows from (21 ) with p = E., and (27)follows from (24)with p = E., = t " l , and o,=r,=O. Using the Stepping-up Lemma (Lemma 16.1), we are now in a position to derive many interesting results from the above corollaries in the case r 2 3 . For the convenience of the reader, we recall that the operation exp is defined recursively by putting
T,
expo (O)= 2 and for any cardinal
;i and
exp, + ( i )= exp"(2')
any integer n.
GROLLARY 17.9. Assume 3.22, p>w, ~ < c( pf ) , and
let
r z 2 he an integer. Then
PROOF. For r = 2 these resultsareidentical to (21)and (23),respectively,and for r > 2 they follow by induction, with the aid of the Stepping-up Lemma. By applying the Stepping-up Lemma to (24), we get
COROLLARY 17.10. Let A, p, o, and T~ be cardinals satisfying i.23, p l w , r , < cf ( p ) , and (T, T , < p , and let r 2 2 be an integer. Then
COROLLARY 17.1 1 . Let p 2 w he a cardinal and r I2 an integer. Then
7*
100
TREES AND POSITIM ORDINARY PARTITION RELATIONS CH.
Iv
and so, a fortiori, (exp,- 1( P ) )+
+
fP + 5 .
(32)
Note that (32) can be directly obtained from the trivial relation P+
+
(P+ );
by induction, using the Stepping-up Lemma. (31) is, however, much stronger, showing that Theorem 17.1 has a special feature not contained in the Steppingu p Lemma. The positive results are in a sense weaker for r > 2 than for r = 2, since all results of the form K+(. . . )' with r 2 3 and K reguiar that we can prove are obtained by repeated applications of the Stepping-up Lemma starting with the case r = 2, except if K has some large cardinal property (for this latter remark cf. e.g. Theorem 29.5). 18. A DIRECT CONSTRUCTION OF THE CANONICAL PARTITION TREE Let K and t be cardinals, fix acoloring f: [ K ] < O + t , and consider the canonical partition tree (K, S, R, 7')associated with f; see the beginning of the proof of Lemma 15.2 for the definition. Simultaneously with this partition tree we also defined a function s: T + K . One can show that s is 1-1and onto and, moreover, that for anyg, h E T w i t h g c h we have s ( g ) s s ( h ) .(We in effect showed the latter assertion in the proof of Lemma 15.2;we omit the simple proof of the former here since we need these facts only in order to give some background to what follows.) So we can define an ordering <, of K which turns K into a tree isomorphic to (7: c ) by putting
5
(1)
and here 5 < / q implies g < q . It is obvious that the tree (K, <,) is just as applicable in proving the existence of large homogeneous sets as is the partition tree ( K, S, R, T). In fact, we shall see that there is some advantage in taking a closer look at the tree (K, if): by so doing we shall be able to prove (a strengthening of) Theorem 17.1 in case b; in the proof we gave in the preceding section we had t o define another partition tree. An important feature in considering the tree (K, <,) is that there is a simple direct definition of the partial ordering
101
A DIRECT CONSTRUCTTON OF THE CANONICAL PARTITION TREE
DEFINITION fg.1. cj Let X G K. Define the equivalence relation a--hxP
-l;
by putting
iff V u ~ [ X ] ~ ~ f ( u u ~ a ~ ) = f ( u u ~ ~( 21) )
for any a, p < K. (ii) For each a < K we define the function g,: a-2 by transfinite recursion on 5
(iii) For any a, P < K put For any a < K, introduce the notation
X u = { p : /3<,a)
= {p
g,(/?)= 1).
Note that we have
Xu.<=X , n 5 ,
(7)
where Xa,
<, so defined. (ii) ( K , <,) is a tree.
LEMMA18.2. (i) I f a < @ and gs(a)= 1, then gsPa=g,. (iii) For any a, B < K and u with B < p and u E [ X , A / ~ ]
(8)
that is, for any u E [ X , u {a)]
PROOF. Ad (i). Assume that gpf t=g,rt holds for some <
gs(t)=g,(5)
(10)
also holds, and the desired result will follow by transfinite induction. We have a --hxp.. according to the assumption gs(a)= 1; so we have a fortiori a =-hxp.
9
since X t , c X#,,. ~ Noting that X#,<= Xu,< in view of (9), this implies that t =hXp,t fl holds just in case 5 =xxm,t a holds; in other words, g # ( ( )= 1 just in case gu(t)=1, which proves (10).
I02
TREES AND POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
Ad (ii). First we prove that <, is a partial ordering, i.e., that it is an irreflexive, antisymmetric, and transitive relation.
(K), (Vc$
(11)
where Stat ( K ) denotes the set of stationary subsets of K. Here is, however, a note of caution: the notational conventions introduced in Section 8 are inadequate for explaining the meaning of this relation. For the proof, let f: [ K ] ~ - +1 i r be a coloring, and consider the tree ( K , <,>. (Strictly speaking, (K, <,) has not been defined, since dom ( f ) is a proper subset of[^]'". So put f'(u)=f(u)whenever u ~ d o m (f), and put f'(u)=Ootherwise, i.e., when u E [K)
holds for any a < K. In fact, assume on the contrary that 1 Y, 12p for some a < K . Then (17.1) implies that there is a 5 < z and a set Z E of order type vt such that f ( { y a } ) = l i ( holds for any ~ E ZIt. follows from (8) that Z u { a ) is a
A DIRECT CONSTRUCTION OF THE CANONICAL PARTlTlON TREE
103
homogeneous set of color 1 45 with respect to f . As Z u {a; has order type v,; 1, this completes the proof of the theorem in case (13) fails for some a < K. Assume therefore that (13) holds for all a < K. Put o = w if p < w, o = p is p is regular, and o = p + if p is singular. Note that o is a regular cardinal and o < K. In fact, since we assumed K > w and p < K, the case o = IC could occur only if p was singular and p + = K; but then L , ( K )Icf (p)< p, which conflicts with our assumption P < L ~ ( K ) = L , ( K( L) = K in the case considered). Put A = {a< K : cf (a)= 0 3 . A is a stationary subset of K as it contains the oth element of an arbitrary club in K. The function
h(a)= sup Y,
is a regressive function on A in view of (13). Fodor’s theorem (Theorem 5.4) implies that there is an ordinal C < K and a set B s A stationary in K such that h(a)=< holds for any a € B. Writing f,(/?)=f({a@) for any a, fl wit?/?
S, r Y,=L in particular
K = Y, for any a E C. We claim that C is a chain of the tree (K, <,).
(14)
In fact, let a, /3 E C with a
X,ny=X,ny
f({daj)= f,(4=
fm=f ( { S P ) - ) ;
the second equality here follows from (14) and (12), since if 6 E Y. = Y,, then &(a) and f,(6) coincide, and if &#Ye= Y, then f , ( S ) = f , ( 6 ) = 0 . Using (8), we obtain that f({6y))=f({6a).) since 6
104
TREES A N 6 POSITIVE ORDINARY PARTITION RELATIONS CH.
Iv
We claim that C is a homogeneous set of color 0. In fact, if /? E C then Ply,= y
(equality holds here by (14)), since Y,c /? (cf. (12)).Let now a, /?E C be such that P
B € xu\Y = xu\y.. Hencethedefinitionof ruin (12)impliesf({a/?))=O;thisshowsthat Cisin fact a homogeneous set of color 0. As C is stationary, this completes the proof of the theorem. The reader may easily see that we even proved slightly more than (11). We illustrate the situation only in a special case when K = I = K2,p = K,, T = K1. The above proof gives e.g. the following: Assume 2N0= K,;let D be a stationary set in K 2 with Ds{a
(15)
and let f: [ D I 2 - 2 be a coloring. Then there is either a set C , G D stationary in K 2 with f " C O = (0) or a set C , of cardinality K, with f"C1 = { 13. Using ideas of J. E. Baumgartner [19763, one can show that this assertion is false if we modify our assumptions by requiring that instead of (15).
Dc{a