andrzej roslanowski and saharon shelah- partition theorems from creatures and idempotent...

8/3/2019 Andrzej Roslanowski and Saharon Shelah- Partition Theorems from Creatures and Idempotent Ultrafilters

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PARTITION THEOREMS FROM CREATURES AND

IDEMPOTENT ULTRAFILTERS

ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

Abstract. We show a general scheme of Ramsey-type results for partitionsof countable sets of finite functions, where one piece is big is interpreted inthe language originating in creature forcing. The heart of our proofs followsGlazers proof of the Hindman Theorem, so we prove the existence of idempo-tent ultrafilters with respect to suitable operation. Then we deduce partitiontheorems related to creature forcings.

0. Introduction

A typical partition theorem asserts that if a set with some structure is dividedinto some number of nice pieces, then one of the pieces is large from the pointof the structure under considerations. Sometimes, the underlying structure is com-plicated and it is not immediately visible that the arguments in hands involve apartition theorem. Such is the case with many forcing arguments. For instance,the proofs of propernes of some forcing notions built according to the scheme ofnorms on possibilities have in their hearts partition theorems stating that at somesituations a homogeneous tree and/or a sequence of creatures determining a con-dition can be found (see, e.g., Roslanowski and Shelah [6, 7], Roslanowski, Shelahand Spinas [8], Kellner and Shelah [5, 4]). A more explicit connection of partitiontheorems with forcing arguments is given in Shelah and Zapletal [9].

The present paper is a contribution to the Ramsey theory in the context of fini-tary creature forcing. We are motivated by earlier papers and notions concerningnorms on possibilities, but we do not look at possible forcing consequences. Thecommon form of our results here is as follows. If a certain family of partial finitefunctions is divided into finitely many pieces, then one of the pieces contains allpartial functions determined by an object (a pure candidate) that can be inter-preted as a forcing condition if we look at the setting from the point of view of thecreature forcing. Sets of partial functions determined by a pure candidate mightbe considered as large sets.

Our main proofs are following the celebrated Glazers proof of the Hindman

Theorem, which reduced the problem to the existence of a relevant ultrafilter on in ZFC. Those arguments were presented by Comfort in [2, Theorem 10.3, p.451]with [2, Lemma 10.1, p.449] as a crucial step (stated here in 2.7). The argumentsof the second section of our paper really resemble Glazers proof. In that sectionwe deal with the easier case of omittorylike creatures (loose FFCC pairs of 1.2(2))

Date: January, 2011.1991 Mathematics Subject Classification. Primary 03E05; Secondary: 03E02, 05D10, 54D80.This research was supported by the United States-Israel Binational Science Foundation. Pub-

lication 957.

1


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2 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

and in the proof of the main conclusion (2.10) we use an ultrafilter idempotent withrespect to operation (defined in 2.4). The third section deals with the case oftight

FFCC pairs of 1.2(4). Here, we consider partitions of some sets of partial functionsall of which have domains being essentially intervals of integers starting with somefixed n < . While the general scheme of the arguments follows the pattern ofthe second section, they are slightly more complicated as they involve sequences ofultrafilters and operations on them. As an application of this method, in 3.9 we givea new proof of a partition theorem by Carlson and Simpson [1, Theorem 6.3]. Thenext section presents a variation of the third section: under weaker assumptionson the involved FFCC pairs we get weaker, yet still interesting partition theorem.Possible applications of this weaker version include a special case of the partitiontheorem by Goldstern and Shelah [3] (see 4.9). These results motivate the fourthsection, where we develop the parallel of the very weak bigness for candidates withlimsup demand on the norms.

Our paper is self-contained and all needed creature terminology is introducedin the first section. We also give there several examples of creating pairs to whichour results may be applied.

Notation: We use standard set-theoretic notation. An integer n is the set {0, 1, . . . , n 1} of all integers smaller than n, and the

set of all integers is called . For integers n < m, the interval [n, m) denotes theset of all integers smaller than m and greater than or equal to n.

All sequences will be indexed by natural numbers and a sequence of objects istypically denoted by a bar above a letter with the convention that x = xi : i < y ,y .

For a set X the family of all subsets of X is denoted by P(X). The domain ofa function f is called dom(f).

An ideal J on is a family of subsets of such that

(i) all finite subsets of belong to J but / J, and(ii) if A B J, then A J and if A, B J then A B J.

For an ideal J, the family of all subsets of that do not belong to J is denoted byJ+, and the filter dual to J is called Jc.

1. Partial creatures

We use the context and notation of Roslanowski and Shelah [6], but below werecall all the required definitions and concepts.

Since we are interested in Ramsey-type theorems and ultrafilters on a countableset of partial functions, we will use pure candidates rather than forcing notionsgenerated by creating pairs. Also, our considerations will be restricted to creatingpairs which are forgetful, smooth ([6, 1.2.5]), monotonic ([6, 5.2.3]), strongly finitary([6, 1.1.3, 3.3.4]) and in some cases omittorylike ([6, 2.1.1]). Therefore we willreformulate our definitions for this restricted context (in particular, val[t] is a setof partial functions), thus we slightly depart from the setting of [6].

Context 1.1. In this paper H is a fixed function defined on and such that H(i)is a finite non-empty set for each i < . The set of all finite non-empty functions fsuch that dom(f) and f(i) H(i) (for all i dom(f)) will be denoted by FH.


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PARTITION THEOREMS FROM CREATURES AND IDEMPOTENT ULTRAFILTERS 3

Definition 1.2. (1) An FP creature1 for H is a tuple

t = (nor, val, dis, mdn, mup) = (nor[t], val[t], dis[t], mtdn, m

tup)

such that nor is a non-negative real number, dis is an arbitrary object and

mtdn < mtup < and

val is a non-empty finite subset ofFH such that dom(f) [mtdn, m

tup)

for all f val.(2) An FFCC pair2 for H is a pair (K, ) such that

(a) K is a countable family of FP creatures for H,(b) for each m < the set Km := {t K : mtup m} is finite and the

set Km := {t K : mtdn m & nor[t] m} is infinite,

(c) is a function with the domain dom() included in the set

{(t0, . . . , tn) : n < , t K and mtup m

t+1dn for < n}

and the range included in P(K) \ {},(d) ift (t0, . . . , tn) then (t K and) m

t0dn = m

tdn < m

tup = m

tnup,

(e) t (t) (for each t K) and(f) ift (t0, . . . , tn) and f val[t], then

dom(f)

{[mtdn, mtup) : n}

and f[mtdn, mtup) val[t] {} for n, and

(g) if t0, . . . , tn dom() and t = t0. . .tn dom(), then

{(s0, . . . , sn) : s (t) for n} (t).

(3) An FFCC pair (K, ) is loose if

(cloose) the domain of is

dom() = {(t0, . . . , tn) : n < , t K and mtup m

t+1dn for < n}.

(4) An FFCC pair (K, ) is tight if(ctight) the domain of is

dom() = {(t0, . . . , tn) : n < , t K and mtup = m

t+1dn for < n},

(ftight) if t (t0, . . . , tn) and f val[t], then f[mtdn, m

tup) val[t] for all

n, and(htight) if s0, s1 K, m

s0up = m

s1dn, f0 val[s0], f1 val[s1] and f = f0 f1,

then there is s (s0, s1) such that f val[s].

Definition 1.3(Cf. [6, Definition 1.2.4])

.Let (K, ) be an FFCC pair for

H.

(1) A pure candidate for(K, ) is a sequence t = tn : n < such that tn K,

mtnup mtn+1dn (for n < ) and limn

nor[tn] = .

A pure candidate t is tight if mtnup = mtn+1dn (for n < ).

The set of all pure candidates for (K, ) is denoted by PC(K, ) and thefamily of all tight pure candidates is called PCtt(K, ).

1FP stands for Forgetful Partial creature2FFCC stands for smoothForgetful monotonic stronglyFinitaryCreature Creating pair


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(2) For pure candidates t, s PC(K, ) we write t s whenever there is asequence un : n < of non-empty finite subsets of satisfying

max(un) < min(un+1) and sn (tun) for all n < .

(3) For a pure candidate t = ti : i < PC(K, ) we define(a) S(t) = {(ti0 , . . . , tin) : i0 < .. . < in < for some n < }, and(b) (t) =

{(s) : s S(t)} and tt(t) =

{(t0, . . . , tn) : n < },

(c) pos(t) =

{val[s] : s (t)} and postt(t) =

{val[s] : s tt(t)},(d) t n = tn+k : k < .

(Above, if s / dom() then we stipulate (s) = .)

Remark 1.4. Loose FFCC and tight FFCC are the two cases of FFCC pairs treatedin this article. The corresponding partition theorems will be slightly different inthe two cases, though there is a parallel. In the loose case we will deal with (t),pos(t) and ultrafilters on the latter set. In the tight case we will use tt(t), postt(t)

and sequences of ultrafilters on postt

(t n) (for n < ).We will require two additional properties from (K, ): weak bigness and weak

additivity (see 1.5, 1.6). Because of the differences in the treatment of the twocases, there are slight differences in the formulation of these properties, so we havetwo variants for each: lvariant and tvariant (where l stands for loose andt stands for tight, of course).

Plainly, PCtt(K, ) PC(K, ), tt(t) (t) and postt(t) pos(t). Also,

if t PCtt(K, ), then t n PCtt(K, ) for all n < .

Definition 1.5. Let (K, ) be an FFCC pair for H and t = ti : i < PC(K, ).

(1) We say that the pair (K, ) has weakladditivity for the candidate t if forsome increasing f : , for every m < we have:

if s0, s1 (t), nor[s0] f(m), ms0dn f(m), nor[s1] f(ms0up) andms1dn > f(m

s0up), then we can find s

(t) such that

msdn m, nor[s] m, and val[s] {f g : f val[s0], g val[s1]}.

(2) The pair (K, ) has weak tadditivity for the candidate t if for some in-creasing f : , for every n,m < we have:if s0 (tn, . . . , tk), k n, nor[s0] f(n + m), s1 (tk+1, . . . , t),nor[s1] f(k + m) and > k, then we can find s (tn, . . . , t) such thatnor[s] m and val[s] {f g : f val[s0], g val[s1]}.

(3) The pair (K, ) has ladditivity if for all s0, s1 K with nor[s0], nor[s1] >1 and ms0up m

s1up there is s (s0, s1) such that

nor[s] min{nor[s0], nor[s1]} 1 and val[s] {f g : f val[s0], g val[s1]}.

The pair (K, ) has tadditivity if for all s0, s1 K with nor[s0], nor[s1] >1 and ms0up = m

s1up there is s (s0, s1) such that

nor[s] min{nor[s0], nor[s1]} 1.

We say that (K, ) has tmultiadditivity if for all s0, . . . , sn K withmsup = m

s+1dn (for < n) there is s (s0, . . . , sn) such that nor[s]

max{nor[s] : n} 1.

Definition 1.6. Let (K, ) be an FFCC pair for H and t = ti : i < PC(K, ).


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(1) We say that the pair (K, ) has weaklbigness for the candidate t wheneverthe following property is satisfied:

()tl if n1, n2, n3 < and pos(t) =

{F : < n1}, then for some s

(t)and < n1 we have

nor[s] n2, msdn n3, and val[s] F.

(2) We say that the pair (K, ) has weaktbigness for the candidate t wheneverthe following property is satisfied:

()tt if n, n1, n2 < and postt(t n) =

{F : < n1}, then for some

s tt(t n) and < n1 we have

nor[s] n2 and val[s] F.

(3) We say that the pair (K, ) has bigness if for every creature t K withnor[t] > 1 and a partition val[t] = F1 F2, there are {1, 2} and s (t)such that nor[s] nor[t] 1 and val[s] F.

Definition 1.7. Let (K, ) be an FFCC pair for H.

(1) (K, ) is simple except omitting if for every (t0, . . . , tn) dom() andt (t0, . . . , tn) for some n we have val[t] val[t].

(2) (K, ) is gluing on a candidate t = ti : i < PC(K, ) if for everyn,m < there are k n and s (tn, . . . , tk) such that nor[s] m.

The following two observations summarize the basic dependencies between thenotions introduced in 1.5, 1.6 separately for the two contexts (see 1.4).

Observation 1.8. Assume (K, ) is a loose FFCC pair, t PC(K, ).

(1) If (K, ) has bigness (ladditivity, respectively), then it has weaklbigness(weak ladditivity, respectively) for the candidate t.

(2) If (K, ) has the weaklbigness for t, k < and pos(t) =


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val[t] ju

H1(j) is such that {f(i) : f val[t]} = A.

For t0, . . . , tn K1 with mtup m

t+1

dn let 1(t0, . . . , tn) consist of all creaturest K1 such that

mtdn = mt0dn, m

tup = m

tnup, u

t =

n

ut , it = it , At At for some n,

and val[t]

f0 . . . fn : (f0, . . . , f n) val[t0] . . . val[tn]

.Also, let 1 be 1 restricted to the set of those tuples (t0, . . . , tn) for which m

tup =

mt+1dn (for < n). Then

(K1, 1) is a loose FFCC pair for H1 with bigness and ladditivity, (K1,

1) is a tight FFCC pair for H1 with bigness and tmultiadditivity,

and it is gluing on every t PCtt(K1, 1).

Example 1.11. Let H2(n) = 2 for n < and let K2 consist of all FP creatures t

for H2 such that = dis[t] [mtdn, m

tup),

= val[t] dis[t]2, nor[t] = log2(|val[t]|).


t+1dn let 2(t0, . . . , tn) consist of all creatures

t K2 such that

mtdn = mt0dn, m

tup = m

tnup, dis[t] = dis[t ], and val[t] val[t] for some

n.

Then (K2, 2) is a loose FFCC pair for H1 which is simple except omitting andhas bigness.

Example 1.12. Let H be as in 1.1 and let K3 consist of all FP creatures t for H

such that = dis[t] [mtdn, m

tup),

val[t] {f FH : dis[t] dom(f) [mtdn, mtup)} satisfies

(g

idis[t]

H(i))(f val[t])(g f),

nor[t] = log957(|dis[t]|).


t+1dn let 3(t0, . . . , tn) consist of all creatures

t K3 such that

mtdn = mt0dn, m

tup = m

tnup, dis[t]

n

dis[t], and

if f val[t], then dom(f)

{[mtdn, mtup) : n} and f[m

tdn, m

tup)

val[t] {} for all n.Also, for t0, . . . , tn K2 with m

tup = m

t+1dn let

3(t0, . . . , tn) consist of all creatures

t K3 such that

mtdn = mt0dn, m

tup = m

tnup, dis[t]

n

dis[t], and

if f val[t], then dom(f)

{[mtdn, mtup) : n} and f[m

tdn, m

tup)

val[t] for all n.

Then

(K3, 3) is a loose FFCC pair for H with bigness and ladditivity,


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(K3, 3) is a tight FFCC pair for H with bigness and tmultiadditivity

and it is gluing on every t PCtt(K3, 3).

Example 1.13. Let N > 0 and HN(n) = N. Let KN consist of all FP creaturest for HN such that

dis[t] = (Xt, t), where Xt [mtdn, mtup), and t : Xt N,

nor[t] = mtup,

val[t] =

f [mtdn,m

tup)N : t f and f is constant on [mtdn, m

tup) \ Xt

.

For t0, . . . , tn KN with mtup = m

t+1dn (for < n) we let N(t0, . . . , tn) consist of

all creatures t KN such that

mtdn = mt0dn, m

tup = m

t0up, Xt0 . . . Xtn Xt,

for each n,either Xt [m

tdn, m

tup) = Xt and t[m

tdn, m

tup) = t ,

or [mtdn

, mtup

) Xt

and t[mt

dn, mt

up) val[t

].

Then

(i) (KN, N) is a tight FFCC pair for HN,(ii) it has the tmultiadditivity and

(iii) it has the weak tbigness and is gluing for every candidate t PCtt(K, ).

Proof. (i) All demands in 1.2(2,4) are easy to verify. For instance, to check1.2(4)(htight) note that:if s0, s1 KN, m

s0up = m

s1dn, f val[s] (for = 0, 1) and s KN is such that

msdn = ms0dn, m

sup = m

s1up, Xs = Xs0[m

s1dn, m

s1up), sXs0 = s0 , s[m

s1dn, m

s1up) = f1,

then s N(s0, s1) and f0 f1 val[s].(ii) The s constructed as in (i) above for s0, s1 will witness the tadditivity aswell. In an analogous way we show also the multiadditivity.(iii) Let t = ti : i < PC

tt(KN, N). Suppose that n, n1, n2 < and

postt(t n) =

{F : < n1}. By the HalesJewett theorem (see [?]) there isk > n2 such that for any partition of

kN into n1 parts there is a combinatorial lineincluded in one of the parts. Then we easily find s N(t0, . . . , tk1) such thatval[s] F for some < n1. Necessarily, nor[s] k 1 n2. This proves theweak tbigness for t. Similarly to (ii) we may argue that (KN, N) is gluing ont.

2. Ultrafilters on loose possibilities

Here we introduce ultrafilters on the (countable) set FH (see 1.1) which contain

sets large from the point of view of pure candidates for a loose FFCC pair. Thenwe use them to derive a partition theorem for this case.

Definition 2.1. Let (K, ) be a loose FFCC pair for H.

(1) For a pure candidate t PC(K, ), we define A0

t= {pos(t n) : n < },

A1t is the collection of all sets A FH such that for some N < wehave

s (t)

nor[s] N & msdn N val[s] A =

,


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A2t is the collection of all sets A FH such that for some N < wehave

(t1 t)(t2 t1)(s (t2))(nor[s] N val[s] A =

.

(2) For < 3 we let uft(K, ) be the family of all ultrafilters D on FH suchthat At D. We also set (for < 3)

uf(K, )def=

{uft(K, ) : t PC(K, )}.

Proposition 2.2. Let(K, ) be a loose FFCC pair for H, t PC(K, ).

(1) A0t A1t A

2t

and hence also uf2t (K, ) uf1t (K, ) uf

0t (K, ).

(2) uf0t (K, ) = .(3) If (K, ) has the weak lbigness for each t t, then uf2t (K, ) is not

empty.(4) If (K, ) has the weaklbigness for t, then uf1t (K, ) = .

(5) Assume CH. Suppose that (K, ) is simple except omitting (see 1.7(1)) andhas the weak lbigness on every candidate t PC(K, ). Then there isD uf2t (K, ) such that

A D

t PC(K, )

pos(t) D & pos(t) A

.

Proof. (2) Note that A0t

has the finite intersection property (fip).

(3) It is enough to show that, assuming (K, ) has the weak lbigness for allt t, A2t has fip. So suppose that for < k we are given a set A A

2t and let

N < be such that

() (t1 t)(t2 t1)(s (t2))(nor[s] N val[s] A =

.

Let N = max{N : < k}. Then we may choose t t such that

() (s

(t

))(nor

[s] N ( < k)(val

[s] A = )

.[Why? Just use repeatedly () for = 0, 1, . . . , k 1; remember t t implies

(t) (t).]For k2 set

F = {f pos(t) : ( < k)(() = 1 f A)}.

Then pos(t) =

{F : k2} and (K, ) has the weak lbigness for t, so we

may use Observation 1.8(2) to pick 0 k2 and s t such that val[sn] F0

for all n < . Consider n < such that nor[sn] > N. It follows from () thatval[sn] A = for all < k. Hence, by the choice of s, 0() = 1 for all < k andtherefore = val[sn]


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Observation 2.3. The sets uft(K, ) (for < 3) are closed subsets of the (Haus-dorff compact topological space) (FH) of non-principal ultrafilters on FH. Hence

each uft(K, ) itself is a compact Hausdorff space.

Definition 2.4. (1) For f FH and A FH we define

f Adef= {g FH : max(dom(f)) < min(dom(g)) and f g A}.

(2) For D1, D2 uf0(K, ) we let

D1 D2def=

A FH : {f FH : (f A) D1} D2

.

Proposition 2.5. (1) If A1, A2 FH and f FH, then

f (A1 A2) = (f A1) (f A2) and{g FH : max(dom(f)) < min(dom(g))} \ (f A1) = f (FH \ A1).

(2) If D1, D2, D3 uf0(K, ), then D1 D2 is a non-principal ultrafilter onFH and D1 (D2 D3) = (D1 D2) D3.

(3) The mapping : uf0(K, ) uf0(K, ) (FH) is right continuous(i.e., for each D1 uf0(K, ) the function uf0(K, ) D2 D1 D2 (FH) is continuous).

Proof. Straightforward, compare with 3.3.

Proposition 2.6. Assume that a loose FFCC pair (K, ) has the weakladditivity(see 1.5(1)) for a candidate t PC(K, ). IfD1, D2 uf

1t (K, ), thenD1 D2

uf1t (K, ).

Proof. Let f : witness the weak ladditivity of (K, ) for t, and let D =D1 D2, D1, D2 uf

1t (K, ). We already know that D is an ultrafilter on FH (by

2.5(2)), so we only need to show that it includes A1t .

Suppose that A A1t and let N < be such that

()1

s (t)

nor[s] N & msdn N val[s] A =

.

Claim 2.6.1. For every s (t), if nor[s] f(N) andmsdn f(N), thenval[s] {f FH : f A D1} = .

Proof of the Claim. Suppose s0 (t), nor[s0] f(N), m

s0dn f(N). Set

B =

{f A : f val[s0]}.

We are going to argue that

()2 B A1t .

So let M = f(ms0up

) + ms0up

+ 957 and suppose s1 (t) is such that nor[s1] Mand ms1dn M. Apply the weak additivity and the choice of M to find s

(t)such that

msdn N, nor[s] N and val[s] {f g : f val[s0] & g val[s1]}.

Then, by ()1, val[s] A = so for some f val[s0] and g val[s1] we have fg A (and max(dom(f)) < min(dom(g))). Thus g (f A) val[s1] B val[s1]and ()2 follows.

Since D1 uf1t (K, ) we conclude from ()2 that B D1 and hence (as val[s0]

is finite) f A D1 for some f val[s0], as desired.


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It follows from 2.6.1 that {f FH : f A D1} A1t and hence (as D2

uf1t (K, )) {f FH : f A D1} D2. Consequently, A D1 D2.

Lemma 2.7 (See [2, Lemma 10.1, p.449]). If X is a non-empty compact Hausdorffspace, an associative binary operation which is continuous from the right (i.e.for eachp X the function q p q is continuous), then there is a idempotentpoint p X (i.e. p p = p).

Corollary 2.8. Assume that a loose FFCC pair (K, ) has weakladditivity andthe weaklbigness for a candidate t PC(K, ). Then

(1) uf1t (K, ) is a non-empty compact Hausdorff space and is an associativeright continuous operation on it.

(2) There is D uf1t (K, ) such that D = D D.

Proof. (1) By 2.2(3), 2.3, 2.5(2,3) and 2.6.

(2) It follows from (1) above that all the assumptions of Lemma 2.7 are satisfiedfor and uf1t (K, ), hence its conclusion holds.

Theorem 2.9. Assume that (K, ) is a loose FFCC pair, t PC(K, ). Let anultrafilterD uf1t (K, ) be such that D D = D. Then

A D

s t

pos(s) A

.

Proof. The main ingredient of our argument is given by the following claim.

Claim 2.9.1. Let(K, ), t andD be as in the assumptions of 2.9. Assume A Dand n < . Then there is s (t) such that

()1 val[s] A, nor[s] n, msdn n, and

()2 (f val[s])(f A D).

Proof of the Claim. Let A := {f FH : f A D} and A := A A. SinceA D = D D we know that A D and thus A D. Hence FH \ A

/ A1t(remember D uf1t (K, )). Therefore, there is s

(t) such that

nor[s] n, msdn n, and val[s] (FH \ A) = .

Then val[s] A and for each f val[s] we have f A D, as desired.

Now suppose A D. By induction on n we choose sn, An so that

(a) A0 = A, An D and An+1 An,(b) sn

(t), nor(sn) n and msnup msn+1dn ,

(c) val[sn] An,(d) iff An+1, then m

snup min(dom(f)),

(e) iff val[sn], then An+1 f An.Suppose we have constructed s0, . . . , sn1 and An so that demands (a)(e) aresatisfied. Set N = m

sn1up + n + 1 (if n = 0 stipulate m

sn1up = 0) and use Claim

2.9.1 to find sn (t) such that

()n1 val[sn] An, nor[sn] N, msndn N, and

()n2 (f val[sn])(f An D).

Put

An+1 := An {g FH : msnup < min(dom(g))}

fval[sn]

f An.


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Since A0t D we know that {g FH : msnup < min(dom(g))} D and since val[sn]

is finite (and by ()n2 ) also

fval

[sn]

f An D. Thus An+1 D. Plainly the other

requirements hold too.After the above construction is carried out we set s = sn : n < . Clearly

s PC(K, ) and s t (remember clause (b)).

Claim 2.9.2. If n0 < .. . < nk < and f val[sn ] for k, thenk

f An0 .

Proof of the Claim. Induction on k. If k = 0 then clause (c) of the choice of sn0gives the conclusion. For the inductive step suppose the claim holds true for kand let n0 < n1 < . . . < nk < nk+1, f val[sn ] (for k + 1). Lettingg = f1 . . . fk+1 we may use the inductive hypothesis to conclude that g An1 .By (a)+(e) we know that An1 An0+1 f0 An0 , so g f0 An0 . Hencef0 g = f0 f1 . . . fk+1 An0 .

It follows from 2.9.2 that pos(s) A (remember (a) above and 1.2(2)(f)).

Conclusion 2.10. Suppose that (K, ) is a loose FFCC pair with weak lbignessand weak ladditivity over t PC(K, ). Assume also that pos(t) is the finiteunion F0 . . . Fn. Then for some i n and s PC(K, ) we have

pos(s) Fi and t s.

Proof. By 2.8 there is D uf1t (K, ) such that D = D D. Clearly for somei n we have Fi D. By 2.9 there is s PC(K, ) such that t s andpos(s) Fi.

3. Ultrafilters on tight possibilities

In this section we carry out for tight FFCC pairs considerations parallel to thatfrom the case of loose FFCC pairs. The main difference now is that we use sequencesof ultrafilters, but many arguments do not change much.

Definition 3.1. Let (K, ) be a tight FFCC pair for H, t = tn : n < PCtt(K, ).

(1) For f postt(t n) let xf = xtf be the unique m > n such that f val[s]

for some s (tn, . . . , tm1). (Note 1.2(4)(ftight).)(2) Iff postt(t n), n < , A FH, then we set

fA = ft A =

g postt(t xf) : f g A

.

(3) We let suft(K, ) be the set of all sequences D = Dn : n < such that

each Dn is a non-principal ultrafilter on postt(t n).(4) The space suft(K, ) is equipped with the (Tichonov) product topology of

n


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(6) For D1, D2 suft(K, ) we define D1 D2 to be a sequence Dn : n <

such that for each n

Dn =

A postt(t n) : setnt (A, D1) D2n

.

Observation 3.2. Let (K, ) be a tight FFCC pair for H and t PCtt(K, ).Suppose f postt(t n), g postt(t xf). Then

(1) f g postt(t n) (note 1.2(4)(htight)) and(2) (f g) A = g (f A) for all A FH.(3) suft(K, ) is a compact Hausdorff topological space. The sets NbA for

A = A0, . . . , An, A postt(t ), n < , form a basis of the

topology of suft(K, ).

Proposition 3.3. Let(K, ) be a tight FFCC pair for H and t PCtt(K, ).

(1) If D1, D2 suft(K, ), then D1 D2 suft(K, ).

(2) The mapping : suft(K, )suft(K, ) suft(K, ) is right continuous.(3) The operation is associative.

Proof. (1) Let D1, D2 suft(K, ), n < , and

Dn =

A postt(t n) : setnt (A, D1) D2n

.

Let A, B postt(t n).(a) If f postt(t n) and A is finite, then f A is finite as well, so it does notbelong to D1xf . Consequently, if A is finite then set

nt

(A, D1) = and A / Dn.

(b) setnt (postt(t n), D1) = postt(t n) D2n (note 3.2(1)). Thus pos

tt(t n) Dn.(c) If A B then setnt (A, D

1) setnt (B, D1) and hence

A B & A Dn B Dn.

(d) setnt (A B, D1) = setnt (A, D

1) setnt (B, D1) and hence

A, B Dn A B Dn.

(e) setnt

(postt(t n) \ A, D1) = postt(t n) \ setnt (A, D1), and hence

A / Dn postt(t n) \ A Dn.

It follows from (a)(e) that Dn is a non-principal ultrafilter on postt(t n) andhence clearly D1 D2 suft(K, ).

(2) Fix D1 suft(K, ) and let A = A : n, A postt(t ).

For n put B = sett(A, D

1) and let B = B : n. Then for eachD2 suf

t(K, ) we have

D1 D2 NbA if and only if D2 NbB.

(3) Let D1, D2, D3 suft(K, ). Suppose n < , A pos(t n). Then

(i) A

(D1 D2) D3n

iff setnt

(A, D1 D2) D3n ifff postt(t n) : f A (D1 D2)xf

D3n, and

(ii) A

D1 (D2 D3)n

iff setnt

(A, D1) (D2 D3)n iff

setnt

setn

t(A, D1), D2

D3n iff

f postt(t n) : f setnt (A, D

1) D2xf

D3n.


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Let us fix f postt(t n) for a moment. Then

f A D1 D2x

f

iff setxft

(f A, D1) D2xf iffg postt(t xf) : g (fA) D1xg

D2xf iff

g postt(t xf) : (f g) A D1xg

D2xf iffg postt(t xf) : (f g) set

nt (A, D

1)

D2xf iff f setnt (A, D

1) D2xf .

Consequently,

f postt(tn) : fA (D1D2)xf

=

f postt(tn) : fsetnt (A, D1) D2xf

and (by (i)+(ii)) A

D1 (D2 D3)n

if and only ifA

(D1 D2) D3n

.

Definition 3.4. Let (K, ) be a tight FFCC pair for H and t PCtt(K, ).

(1) For n < , Bnt is the family of all sets B postt(t n) such that for some

M we have:if s tt(t n) and nor[s] M, then val[s] B = .

(2) suft (K, ) is the family of all D = Dn : n < suft(K, ) such thatBnt

Dn for all n < .

Proposition 3.5. Let(K, ) be a tight FFCC pair for H and t PCtt(K, ).

(1) suft (K, ) is a closed subset of suft(K, ).(2) If (K, ) has the weaktbigness for t, then suft (K, ) = .(3) If (K, ) has the weak tadditivity for t, then suft (K, ) is closed under

.

Proof. (1) Suppose D suft(K, ) \ suft (K, ). Let n < and B B

nt

besuch that B / Dn. Set An = postt(t n) \ B and A = postt(t ) for < n, andlet A = A0, . . . , An. Then D NbA suft(K, ) \ suf

t (K, ).

(2) It is enough to show that, assuming the weak tbigness, each family Bnt hasfip. To this end suppose that B0, . . . , Bm1 Bnt . Pick M0 such that

()

s tt(t n)

< m

nor[s] M0 B val[s] =

.

For m2 set C = {f postt(t n) : ( < m)(f B () = 1)}. By the

weak tbigness we may choose and s tt(t n) such that nor[s] > M0 andval[s] C. Then (by ()) we also have () = 1 and val[s] B for all < m.Hence = val[s]


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have B val[s] = , so for some f val[s0] and g val[s1] we have g f B.Thus val[s1] A = and we easily conclude that A B

mt .]

But D1

suf

t (K, ), so Bmt D

1m and hence, for some f val[s0], we get

fB D1xf . Then f val[s0] setnt

(B, D1).

It follows from 3.5.1 that setnt

(B, D1) Bnt D2n, so B Dn as required.

Corollary 3.6. Assume that(K, ) is a tight FFCC pair with the weaktadditivityand the weak tbigness for t PCtt(K, ). Then there is D suf

t (K, ) such

that D D = D.

Proof. By 2.7+3.2(3)+3.3+3.5.

Theorem 3.7. Assume that (K, ) is a tight FFCC pair, t = tn : n < PCtt(K, ). Suppose also that

(a) D = Dn : n < suft (K, ) is such that D D = D, and

(b) A = An : n < is such that An Dn for all n < .Then there is s = si : i < PC

tt(K, ) such that s t, m

s0dn = m

t0dn and if

i < , si tt(t k), then postt(s i) Ak.

Proof. Let (K, ), t, D and A be as in the assumptions. Then, in particular, Bkt Dk for all k < .

Claim 3.7.1. If M, k < and B Dk, then there is s tt(t k) such thatval[s] B, nor[s] M and (f val[s])(fB Dxf ).

Proof of the Claim. Since D = D D and B Dk, we know that setkt (B, D) Dkand thus B setkt (B, D) Dk. Hence pos

tt(t k) \ (B setkt (B, D)) / Bkt

and we

may find s tt(t k) such that nor[s] M and val[s] B setkt (B, D). This sis as required in the assertion of the claim.

Now we choose inductively si, Bi, ki (for i < ) such that

(i) B0 = A0, k0 = 0,(ii) Bi Dki , Bi Aki , ki < ki+1 < , si (tki , . . . , tki+11),

(iii) val[si] Bi, nor[si] i + 1,(iv) if f val[si], then Bi+1 fBi Dki+1.

Clause (i) determines B0 and k0. Suppose we have already chosen ki and Bi Dki .By 3.7.1 we may find ki+1 > ki and si (tki , . . . , tki+11) such that

nor[si] i + 1, val[si] Bi and (f val[si])(f Bi Dki+1).

We let Bi+1 = Aki+1

{f Bi : f val[si]} Dki+1. One easily verifies therelevant demands in (ii)(iv) for si, Bi+1, ki+1.

After the above construction is carried out, we set s = si : i < . Plainly,s PCtt(K, ), s t and m

s0dn = m

t0dn.

Claim 3.7.2. For each i,k < and s (si, . . . , si+k) we have val[s] Bi.

Proof of the Claim. Induction on k < . If k = 0 then the assertion of the claimfollows from clause (iii) of the choice of si. Assume we have shown the claim fork. Suppose that s (si, . . . , si+k, si+k+1), i < , and f val[s]. Let f0 =f[msidn, m

siup) val[si] and f1 = f[m

si+1dn , m

si+k+1up ) postt(s (i + 1)) (remember

1.2(4)(ftight) and 3.2(1)). By the inductive hypothesis we know that f1 Bi+1, soby clause (iv) of the choice of si we get f1 f0 Bi and thus f = f0 f1 Bi.


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It follows from 3.7.2 that for each i < we have postt(s i) Bi Aki , asrequired.

Conclusion 3.8. Suppose that (K, ) is a tight FFCC pair with weak tbigness andweak tadditivity for t PCtt(K, ).

(a) Assume that, for each n < , kn < and dn : postt(t n) kn. Thenthere is s = si : i < PC

tt(K, ) such that s t, m

s0dn = m

t0dn and for

each n < ,if n is such that si tt(t n), then dnpostt(s i) is constant.

(b) Suppose also that (K, ) has tmultiadditivity. Let dn : postt(t n) k(for n < ), k < . Then there are s = si : i < PC

tt(K, ) and

< k such that s t and for each i < , if n is such that si tt(t n)and f postt(s i), then dn(f) = .

Now we will use 3.8 to give a new proof of CarlsonSimpson Theorem. Thistheorem was used as a crucial lemma in the (inductive) proof of the Dual RamseyTheorem [1, Theorem 2.2].

Theorem 3.9. [Carlson and Simpson [1, Theorem 6.3] ] Suppose that0 < N < ,X =

n


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()vwt for every n,L,M < and a partition F0 . . . FL = postt(t n),

there are i0 = n i1 < i2 i3, L and g0 pos(t[i0, i1)),

g2 pos(t[i2, i3)) and s (ti1 , . . . , ti21) such thatnor[s] M and (g1 val[s])(g0 g1 g2 F).

Observation 4.2. If a tight FFCC pair (K, ) has the weaktbigness for t, thenit has the very weak tbigness for t.

Definition 4.3. Let (K, ) be a tight FFCC pair for H and t PCtt(K, ).

(1) For n < , Cnt is the family of all sets B postt(t n) such that for some

M we have:if i0 = n i1 < i2 i3, g0 pos(t[i0, i1)), g2 pos(t[i2, i3)) ands (ti1 , . . . , ti21), nor[s] M, then B {g0 g1 g2 : g1 val[s]} = .

(2) suft (K, ) is the family of all D = Dn : n < suft(K, ) such that

C

n

t Dn for all n < .Proposition 4.4. Let(K, ) be a tight FFCC pair for H and t PCtt(K, ).

(1) suft (K, ) is a closed subset of suft(K, ), suft (K, ) suf

t (K, ).

(2) If (K, ) has the very weak tbigness for t, then suft (K, ) = .(3) If D suft (K, ), n < and B C

nt , then set

nt (B, D) = pos

tt(t n).(4) suft (K, ) is closed under the operation (defined in 3.1(6)).

Proof. (1) Since in 4.3(1) we allow i1 = i0 and i3 = i2 (so g0 = g2 = ), we easilysee that Cn

t Bn

t. Hence suft (K, ) suf

t (K, ). The proof that suf

t (K, ) is

closed is the same as for 3.5(1).(2) Like 3.5(2).(3) Let M be such that B {g0 g1 g2 : g1 val[s]} = whenever g0

pos(t[n, i1)), g2 pos(t[i2, i3)), s (t[i1, i2)), nor[s] M, n i1 < i2 i3.We will show that this M witnesses f B Cxft

for all f postt(t n).So suppose that f postt(t n) and xf i1 < i2 i3, g0 pos(t[xf, i1)),

s (t[i1, i2)), nor[s] M and g2 pos(t[i2, i3)). Then f g0 pos(t[n, i1))(remember 3.2(1)) and consequently (by the choice of M) B

(f g0) g1 g2 :

g1 val[s]

= . Let g1 val[s] be such that f g0 g1 g2 B. Then

g0 g1 g2 fB witnessing that (f B)

g0 g1 g2 : g1 val[s]

= .

Since Cxft

Dxf we conclude now that fB Dxf so f setnt (B, D).

(4) Suppose D1, D2 suft (K, ), D = D1 D2. Let B C

nt

, n < . By (3)we know that setnt (B, D1) = pos

tt(t n) D2n and thus B Dn. Consequently,Cnt Dn for all n < , so D suf

t (K, ).

Corollary 4.5. Assume that (K, ) is a tight FFCC pair with the very weak tbigness for t PCtt(K, ). Then there is D suf

t (K, ) such that D D = D.

Theorem 4.6. Assume that (K, ) is a tight FFCC pair for H, t = tn : n < PCtt(K, ). Let D suf

t (K, ) be such that D D = D and suppose that

An Dn for n < . Then there are sequences ni : i < , g3i, g3i+2 : i < ands3i+1 : i < such that for every i < :

() 0 = n0 n3i n3i+1 < n3i+2 n3i+3 < ,() if j = 3i or j = 3i + 2, then gj pos(t[nj, . . . , nj+1)),() if j = 3i + 1, then sj (tnj , . . . , tnj+11) and nor[sj] j,


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() if g3+1 val[s3+1] for [i, k), i < k, then3k1j=3i

gj An3i .

Proof. Parallel to 3.7, just instead of val[si] use {gi1 g gi+1 : g val[si]}.

Conclusion 4.7. Assume that (K, ) is a tight FFCC pair with the very weak tbigness for t PCtt(K, ). Suppose that for each n < we are given kn < and a mapping dn : postt(t n) kn. Then there are sequences ni : i < ,g3i, g3i+2 : i < , s3i+1 : i < and ci : i < such that for each i < :

() 0 = n0 n3i n3i+1 < n3i+2 n3i+3 < , ci kn3i ,() if j = 3i or j = 3i + 2, then gj pos(t[nj, . . . , nj+1)),() if j = 3i + 1, then sj (tnj , . . . , tnj+11) and nor[sj ] j,() if i < k and f pos(t[n3i, n3k)) are such that

g3 g3+2 f and f[ms3+1dn , m

s3+1up ) val[s3+1] for all [i, k),

then dn3i(f) = ci.Example 4.8. Let (G, ) be a finite group. For a function f : S G and a Gwe define a f : S G by (a f)(x) = a f(x) for x S. Let HG(m) = G (form < ) and let KG consist of all FP creatures t for HG such that

nor[t] = mtup, dis[t] = ,

val[t] [mtdn,m

tup)G is such that (f val[t])(a G)(a f val[t]).

For t0, . . . , tn KG with mt+1dn = m

tup (for < n) we let G(t0, . . . , tn) consist of

all creatures t KG such that

mtdn = mt0dn, m

tup = m

tnup,

val[t] {f [mtdn,m

tup)G : ( n)(f[mtdn, m

tup) val[t])}.

Then

(1) (KG, G) is a tight FFCC pair for HG.(2) If |G| = 2, then (KG, G) has the very weak tbigness for every candidate

t PCtt(KG, G).

Proof. (1) Straightforward.(2) Let G = ({1, 1}, ). Suppose that t PCtt(KG, G) and pos

tt(t n) =F0 . . . FL, n, L, M < . For future use we will show slightly more than neededfor the very weak bigness.

We say that N n + M is good (for L) if

() there are j2 j1 > N, g0 pos(t[n, N)), g2 pos(t[j1, j2)) and s G(t[N, j1)) such that {g0 g1 g2 : g1 val[s]} F.

(Note that if s is as in (), then also nor[s] = mtj1dn j1 > N M.) We are

going to argue that() almost every N n + M is good for some L.

So suppose that () fails and we have an increasing sequence n + M < N(0)

andrzej roslanowski and saharon shelah- partition theorems from creatures and idempotent...

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