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SOME RESULTS IN POLYCHROMATIC RAMSEY THEORY

URI ABRAHAM, JAMES CUMMINGS, AND CLIFFORD SMYTH

1. Introduction

Classical Ramsey theory (at least in its simplest form) is concerned with problemsof the following kind: given a set X and a colouring of the set [X]n of unorderedn-tuples from X, find a subset Y ⊆ X such that all n-tuples in [Y ]n get the samecolour. Subsets with this property are called monochromatic or homogeneous, and atypical positive result in Ramsey theory has the form that when X is large enoughand the number of colours is small enough we can expect to find reasonably largemonochromatic sets.

Polychromatic Ramsey theory is concerned with a “dual” problem, in which weare given a colouring of [X]n and are looking for subsets Y ⊆ X such that any twodistinct n-tuples in [Y ]n get different colours. Subsets with this property are calledpolychromatic or rainbow. Naturally if we are looking for rainbow subsets then ourtask becomes easier when there are many colours. In particular given an integer kwe say that a colouring is k-bounded when each colour is used for at most k manyn-tuples.

At this point it will be convenient to introduce a compact notation for statingresults in polychromatic Ramsey theory. We recall that in classical Ramsey theorywe write κ → (α)nk to mean “every colouring of the n-tuples from κ in k colourshas a monochromatic set of order type α”. We will write κ→poly (α)nk−bd to mean“every k-bounded colouring of the n-tuples from κ has a polychromatic set of ordertype α”. We note that when κ is infinite and k is finite a k-bounded colouring willuse exactly κ colours, so we may as well assume that κ is the set of colours used.

Polychromatic Ramsey theory in the finite case has been extensively studiedby finite combinatorists [?, ?, ?, ?], sometimes under the name “Rainbow Ramseytheory” or “Sub-Ramsey theory”. In particular the quantity sr(Kn, k), which in ournotation is the least m such that m→poly (n)2

k−bd, has been investigated; it growsmuch more slowly than the corresponding classical Ramsey number, for example itis known [?] that sr(Kn, k) ≤ 1

4n(n− 1)(n− 2)(k− 1) + 3.In this paper we investigate polychromatic versions of some classic results in in-

finite Ramsey theory. Interestingly we will also often find that, as in the finite case,polychromatic Ramsey numbers grow more slowly than their classical counterparts.

• The polychromatic Ramsey theory of the infinite was first investigated byGalvin [?]. In Section 2 we sketch a couple of results which motivated ourwork in the area.

2000 Mathematics Subject Classification. Primary 03E02, 03E35, 03E50; Secondary 05D10.Key words and phrases. Polychromatic Ramsey theory, forcing axioms.Second author partially supported by NSF Grant DMS-0400982.

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2 URI ABRAHAM, JAMES CUMMINGS, AND CLIFFORD SMYTH

We describe some machinery (dual colourings) for translating positiveresults in monochromatic Ramsey theory into positive results in polychro-matic Ramsey theory. We use dual colourings to show that if κ → (λ)nkthen κ →poly (λ)nk−bd. In particular the classical infinite Ramsey theo-rem implies that ω →poly (ω)nk−bd for all n and k. We then show thatassuming the Continuum Hypothesis (CH) we have the negative relationω1 6→poly (ω1)2

2−bd.• It is natural to ask whether CH was necessary to show ω1 6→poly (ω1)2

2−bd.We recall that Todorcevic [?] showed in ZFC that ω1 6→ [ω1]2ω1

, which givesa very strong counterexample to ω1 → (ω1)2

2.The polychromatic situation is quite different. Todorcevic [?] proved a

result which implies that, assuming the Proper Forcing Axiom (PFA), wehave the positive relation ω1 →poly (ω1)2

2−bd. In Section 3 we will give analternative proof of this implication from PFA; Todorcevic’s methods givethe consistency of ω1 →poly (ω1)2

2−bd relative to the consistency of ZFC, itis not clear that the methods of Section 3 will do this. See the discussionat the end of Section 3.• Since PFA implies that ω1 →poly (ω1)2

2−bd, it is natural to ask whetherMartin’s Axiom (MA) suffices for this partition relation. In Section 4 weshow that this relation is in fact independent of MA. The proof buildson ideas of Abraham and Todorcevic [?], and shows that we may forcethe existence of a 2-bounded colouring of [ω1]2 which has no uncountablepolychromatic set in any ccc forcing extension.

• After hearing the results of Section 4, Sy Friedman asked for a concreteexample of a 2-bounded colouring of [ω1]2 which has no uncountable poly-chromatic set, but which acquires one in some ccc forcing extension. InSection 5 we construct such a colouring from the assumption that CH holdsand there is a Souslin tree.• The argument of Section 3 actually shows that under PFA every 2-bounded

colouring of pairs from ω1 has a stationary polychromatic set. So a 2-bounded colouring of pairs from ω2 has many polychromatic sets of ordertype ω1 which are stationary in their supremum. In Section 6 we show thatunder MM a 2-bounded colouring of pairs from ω2 has a polychromatic setof order type ω1 which is club in its supremum.

• In Section 7 we show that PFA does not suffice for the result of Section 6.We do this by isolating a consequence of square which is strong enough tobuild a 2-bounded colouring of pairs from ω2 with no polychromatic closedcopy of ω1, yet weak enough to be consistent with PFA.

• Finally in Section 8 we prove some positive partition results from GCHassumptions, which are stronger than the easy results obtained from theErdos-Rado theorem by the dual colouring trick. The results give a fairlyclear picture of which polychromatic partition relations are true underGCH, at least at successors of regular cardinals.

Notation: our notation is standard. One point is worthy of comment, namely thatwhen f is a colouring of pairs of ordinals from some set X and α, β ∈ X with α < βwe will typically write “f(α, β)” instead of the more correct “f({α, β})”. Wheneverwe use this convention it is safe to assume that the first argument of f is less thanthe second argument.

SOME RESULTS IN POLYCHROMATIC RAMSEY THEORY 3

Directions for further research:

(1) It follows from the results in Section 8 that under CH we have the relationω2 →poly (α)2

ω−bd for every α < ω2. In a projected sequel to this paper weshow that a wide range of possibilities is consistent with not-CH; in partic-ular each of the relations ω2 →poly (ω1 + 1)2

ω−bd and ω2 6→poly (ω1)22−bd is

consistent with 2ω = ω2.(2) Jindrich Zapletal and Otmar Spinas pointed out that polychromatic Ram-

sey theory and classical Ramsey theory both fall under the umbrella of“canonical Ramsey theory”, in which we are given an arbitrary colouringof [X]n and look for a large Y ⊆ X so that the colouring is “simple” on[Y ]n. One way to look at the results of this paper is to see them as frag-ments of a canonical Ramsey theory for [ω1]2 and [ω2]2; such a theory willclearly be fraught with independence results. On a related note it may beinteresting to explore connections with “structural Ramsey theory”.

(3) It is natural to ask for a polychromatic analogue of the Galvin-Prikry the-orem. Otmar Spinas pointed out that such a result follows easily from acanonical colouring result of Promel and Voigt [?]. To be precise if we aregiven a 2-bounded Borel colouring of [ω]ω then there is an infinite P ⊆ ωsuch that the colouring is 1-1 on [P ]ω.

(4) Many natural questions remain open, we give a sampling:(a) Construct in ZFC a counterexample to ω2 →poly (ω2)2

2−bd.(b) What can we say about colourings of [ω1]3? See the remarks after

Theorem 1.(c) Suppose that every 2-bounded colouring of [ω2]2 has an uncountable

polychromatic set. Must every 2-bounded colouring of [ω1]2 have anuncountable polychromatic set?

(d) Under what circumstances can “ccc indestructibly bad” colourings ofthe sort constructed in Section 4 exist? Are they compatible with CH?Do they exist in L?

(e) For which inaccessible κ does κ→poly (κ)22−bd hold? In particular does

this partition relation characterise the weakly compact inaccessiblecardinals?

(f) The GCH results of Section 8 only work at successors of regular car-dinals. What happens at successors of singular cardinals?

Acknowledgments: We would like to thank Mirna Dzamonja, Matt Foreman, SyFriedman, Peter Lumsdaine, Lajos Soukup, Otmar Spinas, Stevo Todorcevic, andJindra Zapletal for their helpful comments and suggestions.

2. Results of Galvin

In this section we sketch the proofs of some basic results due to Galvin [?].

2.1. Dual colourings.

Definition 1. Let X be a set and let f be a k-bounded colouring of [X]n. We fixa linear ordering ≺ of [X]n and define the dual colouring f∗ : [X]n → k as follows:if f(a) = c then enumerate the n-tuples with colour c in ≺-increasing order asa0 ≺ a1 ≺ . . . aj, and define f∗(a) = i where a = ai.


It is easy to see that if X is monochromatic for f∗ then it is polychromatic forf . It follows immediately that

κ→ (λ)nk =⇒ κ→poly (λ)nk−bd.

So every theorem in classical Ramsey theorem implies a corresponding theorem inpolychromatic Ramsey theory. In particular we recall that Baumgartner and Hajnal[?] showed that ω1 → (α)2

2 for every countable ordinal α. It follows immediatelythat ω1 →poly (α)2

2−bd for every countable α.The following result shows that in general we can not improve on this; the proof

follows a standard paradigm for building “wild” colourings using cardinal arithmeticassumptions.

2.2. A CH example.

Theorem 1. Let CH hold. Then there is a 2-bounded colouring of [ω1]2 such thatfor all uncountable X ⊆ ω1 there are uncountably many colours which appear twiceon pairs in [X]2. In particular ω1 6→poly (ω1)2

2−bd under CH.

Proof. Using CH enumerate [ω1]ω as 〈Aα : α < ω1〉. Now we build the colouringinductively, at step γ assigning colours to pairs {β, γ} for β < γ. We assume thatthe colouring of [γ]2 which we have built when we reach stage γ is 2-bounded.

We enumerate the sets Aα where Aα ⊆ γ and α < γ as 〈Bj : j < ω〉. We alsochoose colours 〈ck : k < ω〉 which were not yet used. Now we choose the first twopoints in B0 and colour the pairs they form with γ using colour c0, take the first twopoints not yet coloured in B1 and colour the pairs they form with γ using c1 andso on. If at the end there are points α < γ such that {α, γ} was not yet colouredthen enumerate these pairs and colour them inductively giving each one a colournot yet used.

To see that this works let X be unbounded in ω1, and let Y = Aα be the initialsegment of X consisting of the first ω points. Let γ ∈ X with γ > α, sup(Y ). Thenby construction there exist two points ρ and σ in Y such that the pairs {ρ, γ} and{σ, γ} are given the same colour. ¤

Remark 1. Matt Foreman pointed out that the theorem of Baumgartner and Hajnal[?] used above is specific to colourings of pairs: if we fix injections fγ : γ ↪→ ωfor each countable γ and then colour {α, β, γ}< red or blue according to whetherfγ(α) < fγ(β) or fγ(α) > fγ(β), then there is no red-homogeneous set of typeω+ 2 and no blue-homogeneous set of type ω+ 1. It is unclear what is the situationregarding the relation ω1 →poly (α)3

2−bd for α ≤ ω1.

Remark 2. Mirna Dzamonja pointed out that the use of CH in the argumentabove is an overkill, and that much weaker guessing principles consistent with largecontinuum will do. For example the stick principle asserts that there is X ⊆ [ω1]ω

such that |X | = ω1 and for every Y ∈ [ω1]ω1 there is A ∈ X ∩ [Y ]ω; this is exactlythe consequence of CH used in the proof of Theorem 1.

Remark 3. Theorem 1 has an easy generalisation to higher cardinals, which weleave to the reader. See Section 8 of this paper for some positive results from GCHassumptions which do not follow trivially from the Erdos-Rado theorem.


3. A positive result for ω1 from PFA

In this section we will give an alternative proof of a result by Todorcevic [?]which implies that ω1 →poly (ω1)2

2−bd under PFA. The result is more general thanthis in several ways, to state it we introduce the following definition:

Definition 2. Let κ be a cardinal. A colouring of [X]n is (< κ)-bounded if andonly if each colour is used for fewer than κ many n-tuples.

Of course our previous notion of k-boundedness is just the special case withκ = k + 1.

Theorem 2 (Todorcevic [?]). (PFA) If f is a (< ω)-bounded colouring of [ω1]2

then ω1 is the union of ω many polychromatic sets.

The proof will occupy most of this section. We will allow ourselves severaldigressions.

3.1. Normal colourings. We introduce a special property of colourings (normal-ity), and show that we can sometimes decompose a colouring into a small numberof normal colourings.

Definition 3. Let X be a set of ordinals. A colouring f of [X]2 is normal if andonly if for all pairs a = {α, β}< and b = {γ, δ}<, if f(a) = f(b) then β = δ.

Lemma 1. Let λ be an infinite cardinal and let κ = λ+. If f is a (< κ)-boundedcolouring of [κ]2 then there exist sets Xj for j < λ such that κ =

⋃j Xj and

f ¹ [Xj ]2 is normal.

We prove Lemma 1 using elementary substructures. This is an overkill for thetask at hand, but introduces ideas which will be crucial at several points later inthe paper.

Proof. Let θ be a large regular cardinal, let <θ be a well-ordering of Hθ and letA = (Hθ, <θ). Build an increasing continuous chain 〈Ni : i < κ〉 such that

• κ, λ, f ∈ N0.• Ni ≺ A with |Ni| = λ and Ni ∩ κ ∈ κ.

Let δi = Ni ∩ κ so that {δi} is club in κ. Let a = {α, β}< and let i be such thatδi ≤ β < δi+1.

Let X be the set of pairs b with f(b) = f(a), then X is definable in A from fand a so easily X ∈ Ni+1. Since f is (< κ)-bounded, |X| < κ and so µ =def |X| ∈Ni+1 ∩ κ, hence µ ⊆ Ni+1 since Ni+1 ∩ κ ∈ κ.

Let g be the <θ-least bijection between µ and X, then g ∈ Ni+1 so X = g[µ] ⊆Ni+1. That is to say every pair b with f(b) = f(a) lies in [δi+1]2.

We may now easily decompose κ as⋃j<λXj where |Xj ∩ [δi, δi+1)| ≤ 1 for all

i, j. It follows immediately from the discussion above that f ¹ [Xj ]2 is normal. ¤

3.2. Proper forcing. We give a brief review of Shelah’s theory of proper forcing,referring the reader to the survey by Abraham [?] for the details.

Definition 4. Let P be a forcing poset and let θ be a regular cardinal so large thatP (P) ∈ Hθ. Let N ≺ Hθ be countable with P ∈ N . Then q ∈ P is (N,P)-generic ifand only if for every dense subset D ⊆ P with D ∈ N , D ∩N is predense below q.


This can be motivated as follows: if π : N ' N is the Mostowski collapseisomorphism from N to a transitive set N , then p is an (N,P)-generic condition ifand only if it forces that π[GP ∩N ] is an N -generic filter on π(P). We note that ingeneral π[GP ∩N ] will not lie in V .

Definition 5. Let P be a forcing poset and let θ be a regular cardinal so large thatP (P) ∈ Hθ. Let N ≺ Hθ be countable with P ∈ N . Then q ∈ P is (N,P)-stronglygeneric if and only if for every dense subset D ⊆ P with D ∈ N , there is r ∈ D∩Nwith q ≤ r.

A strongly generic condition is one which not only forces that π[GP ∩ N ] isN -generic, but actually determines π[GP ∩ N ]. Strongly generic conditions occurnaturally in the study of forcing posets which add no new reals.

We will find several uses for the following standard facts about elementary sub-structures and generic conditions:

Lemma 2. Let N ≺ Hθ be countable and let δ = N ∩ ω1.Then(1) If C ∈ N is a club subset of ω1, then δ ∈ C.(2) If S ∈ N is a subset of ω1 and δ ∈ S, then S is stationary.(3) Let P be a forcing poset with P (P) ∈ Hθ and suppose that P ∈ N . Then

(a) For any P-generic G, Hθ[G] = HV [G]θ and N [G] =def {τG : τ ∈ N} ≺

Hθ[G].(b) p ∈ P is P-generic if and only if it satisfies either of the following

equivalent conditions:(i) p ° N [G] ∩On = N ∩On.

(ii) p ° N [G] ∩ V = N ∩ V .(c) If p ∈ P is P-generic and C ∈ N is a P-name for a club subset of ω1

then p ° δ ∈ C.

Definition 6. Let X be uncountable.(1) A set C of countable subsets of X is club if and only if every countable

subset of X is contained in an element of C, and C is closed under unionsof chains of length ω.

(2) A set S of countable subsets of X is stationary if and only if S meets everyclub set C.

The following facts are standard:(1) Let S be a set of countable subsets of the uncountable set X.

(a) S is stationary if and only if for every F : <ωX → X there is annonempty set A ∈ S with F [<ωA] ⊆ A.

(b) S is stationary if and only if for every countable first order language Land every L-structure X with underlying set X, there is A ∈ S withA ≺ X .

(c) (Fodor’s lemma) If S is stationary and F : S → X is regressive in thesense that F (A) ∈ A for all A ∈ S, then there is T ⊆ S stationarywith F ¹ T constant.

(2) If Y and Z are uncountable sets with Y ⊆ Z then(a) If S is a stationary set of countable subsets of Y , then the set of

countable B ⊆ Z with B ∩ Y ∈ S is a stationary set of countablesubsets of Z.


(b) If T is a stationary set of countable subsets of Z, then {B∩Y : B ∈ T}is a stationary set of countable subsets of Y .

Definition 7. Let P be a forcing poset. Then P is proper if and only if for ev-ery uncountable set X and every stationary set S of countable subsets of X, thestationarity of S is preserved by P.

Properness can be characterised in terms of generic conditions: P is proper ifand only if for some (equivalently all large) θ such that P (P) ∈ Hθ, there is a clubset of countable N ≺ Hθ such that for all p ∈ P ∩ N there is q ≤ p where q is(N,P)-generic.

All ccc and all countably closed forcing posets are proper. The standard posetsfor adding new reals (Sacks, Laver, Mathias, Miller et cetera) are all proper. Proper-ness is preserved by countable support iterations.

The Proper Forcing Axiom (PFA) is the assertion that for every proper P andevery family F of dense subsets of P with |F| = ω1, there is a filter F on P suchthat F ∩D 6= ∅ for all D ∈ F .

3.3. Amalgamating finite polychromatic sets. Let f be a normal and (< ω)-bounded colouring of [ω1]2. We consider the problem of amalgamating finite poly-chromatic sets, that is finding conditions under which the union of two such sets ispolychromatic. This sort of “amalgamation problem” is ubiquitous in forcing, andalmost always arises when we are trying to prove that a forcing poset is proper: af-ter all the very definition of generic condition demands that we should amalgamateextensions of the generic condition p with certain conditions lying in the elementarysubstructure N .

If X is a finite subset of ω1 we define

F (X) =def {α : ∃γ ∃b ∈ [X]2 f(α, γ) = f(b)}Since f is normal, F (X) is the set of α such that there exist β, γ ∈ X with α, β < γand f(α, γ) = f(β, γ). Clearly F (X) ⊆ max(X), and since f is (< ω)-bounded wesee that F (X) is finite.

When X and Y are finite sets of ordinals we will abuse notation slightly bywriting “X < Y ” as a shorthand for “max(X) < min(Y )”. Similarly when ζ is anordinal, “X < ζ” and “ζ < X” have the obvious meanings.

Lemma 3. Let X < α < Y where X ∪ {α} and X ∪ Y are both polychromatic forf . If α /∈ F (X ∪ Y ) then X ∪ {α} ∪ Y is polychromatic for f .

Proof. Since f is normal, the only problem that can occur is that there may existδ, ε, ζ ∈ X ∪ {α} ∪ Y with δ < ε < ζ and f(δ, ζ) = f(ε, ζ).

Since X ∪{α} is polychromatic, it must be that ζ ∈ Y . Since X ∪Y is polychro-matic, either δ or ε equals α. But both of these possibilities are ruled out by theassumption that α /∈ F (X ∪ Y ). ¤

Now we generalise this lemma somewhat. Say that a finite partial function fromω1 to ω is good for f if and only if for all n the set {i ∈ dom(p) : p(i) = n}is polychromatic. Good partial functions will eventually be building blocks in aforcing poset for decomposing ω1 into ω many polychromatic sets.

The following lemma follows immediately from Lemma 3.

Lemma 4. Let p, q, r be finite partial functions from ω1 to ω such that


(1) dom(p) < dom(q) < dom(r).(2) q is 1-1.(3) p ∪ q and p ∪ r are both good for f .(4) dom(q) is disjoint from F (dom(p) ∪ dom(r)).

Then p ∪ q ∪ r is good for f .

3.4. Elementary substructures. We need some standard facts about elementarysubstructures M ≺ Hθ where M is countable and θ > ω1 is regular.

Firstly if x ∈ M is countable then x ⊆ M . In particular M ∩ ω1 is an initialsegment of ω1, so M ∩ ω1 = δ for some δ ∈ ω1.

Secondly suppose that λ ∈M is a regular cardinal. Then Hλ ∈M , and so also aset of Skolem functions for Hλ is in M . It follows that M ∩Hλ ≺ Hλ. In particularif λ is definable in Hθ then λ ∈M for all countable M ≺ Hθ, and so M ∩Hλ ≺ Hλ

for all such M .Thirdly let M,N ≺ Hθ be countable with M ∈ N . Then as we saw above

M ⊆ N , so in particular M ≺ N . Also M and ω1 are both in N so M ∩ ω1 ∈ N ,in particular M ∩ ω1 < N ∩ ω1.

In the following section we will define a forcing poset in which each conditionis partially composed of countable elementary substructures of Hω2 . The idea isthat these models will act as “side conditions” which will avoid some obstacle toproperness; the idea of using models as side conditions in proper forcing is due toTodorcevic [?].

As motivation for the use of countable elementary substructures of Hω2 , we provea lemma about the kind of amalgamation problem discussed in the last section.

Definition 8. A sequence ~p = 〈pα : α < ω1〉 of finite partial functions from ω1 toω is increasing if and only if α < β =⇒ dom(pα) < dom(pβ) for all α, β.

We note that by an easy induction, if ~p is increasing then α ≤ min dom(pα).Also easily every such sequence ~p is a member of Hω2 . We recall that f is a normal(< ω)-bounded colouring of [ω1]2 and that for finite X ⊆ ω1, F (X) is the set of αwith f(α, γ) = f(β, γ) for some β, γ ∈ X. The following technical lemma will beused to show that the main forcing for Theorem 2 is proper.

Lemma 5. Let M ≺ Hω2 be countable with f ∈M , and let δ = M ∩ω1. Let p andq be finite partial functions from ω1 to ω such that

(1) dom(p) ⊆ δ, dom(q) ⊆ ω1 \ δ (so in particular dom(p) < dom(q)).(2) p ∪ q is good for f .(3) q is 1-1.

Let ~r be an increasing ω1-sequence of partial functions from ω1 to ω such that(1) ~r ∈M .(2) For all i < ω1, p ∪ ri is good for f .

Then there exists S ∈M a stationary subset of ω1 such that for all large enoughi ∈ S, p ∪ q ∪ ri is good for f .

Proof. Consider the function g on limit ordinals α < ω1 given by

g : α 7→ max{max dom(p),max(F (dom(p) ∪ dom(rα)) ∩ α)}The function g is in M because f, F,~r ∈ M . By Fodor’s lemma and elementaritythere exist S ∈M stationary and η ∈ δ such that g is constant on S with value η.We note that dom(p) ⊆ η.


If i ∈ S and dom(q) < i then we see that(1) dom(p) < dom(q) < dom(ri).(2) η < dom(q) < i.(3) F (dom(p) ∪ dom(ri)) ∩ i ⊆ η.

So dom(q) is disjoint from F (dom(p) ∪ dom(ri)), and since also q is 1-1 we mayapply Lemma 4 to conclude that p ∪ q ∪ ri is good for f . ¤

3.5. The forcing poset Pf . Recall that f is a normal (< ω)-bounded colouringof [ω1]2, and that a finite partial function q from ω1 to ω is good for f if and onlyif {i : q(i) = n} is polychromatic for each n. We define a forcing poset Pf whosegoal is to add a decomposition of ω1 into ω polychromatic sets. The conditions arepairs (p,M) where

(1) p is a finite partial function p from ω1 to ω, and p is good for f .(2) M is a nonempty finite set {M0, . . .Mn−1} where Mi ≺ Hω2 , Mi is count-

able and Mi ∈Mi+1 for i+ 1 < n.(3) For all α < β < ω1, if p(α) = p(β) then there exists i such that α <

Mi ∩ ω1 ≤ β.The ordering is by extension, formally (p,M) ≤ (q,N ) if and only if p ⊇ q and

M⊇ N .We record a couple of remarks about the poset Pf :(1) Every condition is an element of Hω2 .(2) If s = (p,M) is a condition with M = {M0, . . .Mn−1} and we define

s ¹Mi = (p ¹Mi,M∩Mi)

then s ¹Mi is a condition and s ¹Mi ∈Mi and s ≤ s ¹Mi.If γ < ω1 then we say that the condition (p,M) is below γ if and only if dom(p) <

γ and M ∩ ω1 < γ for all M ∈ M. If (p,M) ∈ N for some countable N ≺ Hω2

with N ∩ ω1 = γ then easily (p,M) is below γ. The converse is however false ingeneral, if (p,M) is below N ∩ ω1 then p ∈ N but the models in M may not lie inN .

The following lemma shows that by forcing with Pf we add a total function fromω1 to ω.

Lemma 6. For every α < ω1, the set of conditions (p,M) with α ∈ dom(p) isdense.

Proof. Suppose that (p,M) is a condition and that α /∈ dom(p). Choose n ∈ ωwith n /∈ rge(p) and extend p to q with dom(q) = dom(p) ∪ {α}, q(α) = n. Then(q,M) is as required. ¤

It is also helpful to know that densely many conditions have a model in the“M-part” which contains the whole of the “p-part”.

Lemma 7. If (p,M) is a condition and N ≺ Hω2 is countable with (p,M) ∈ N ,then (p,M∪ {N}) is a condition extending (p,M).

The next two lemmas are the motivation for clause 3 in the definition of theconditions.

Lemma 8. Let (p,M) be a condition and δj = Mj ∩ ω1 for Mj ∈M. Then(1) p ¹ δ0 is 1-1.


(2) p ¹ [δi, δi+1) is 1-1 for all i.(3) If Mn is the largest model in M then p ¹ ω1 \ δn is 1-1.

Proof. We only prove the second claim, the others are similar. Suppose that δi ≤α < β < δi+1. Then there is no j such that α < δj ≤ β, so that p(α) 6= p(β). ¤

Lemma 9. Let (p,M) be a condition where M = {M0, . . .Mn−1} say. Let δj =Mj ∩ ω1 for each j, and let q = p ¹Mk for some k < n. If ~r ∈Mk is an increasingω1-sequence of finite partial functions ri such that q ∪ ri is good for every i < ω1,then there is a stationary set S ∈Mn−1 such that p ∪ ri is good for all i ∈ S.

Proof. Apply Lemma 5 repeatedly with p ¹ δj , p ¹ [δj , δj+1), and Mj playing theroles of p, q,M for j = k, k + 1, . . . n− 1 ¤

3.6. Properness of Pf . Now we are ready to argue that Pf is proper. Let θ bea large regular cardinal, and let s = (p,M) ∈ N ≺ Hθ where N is countable. LetM = N ∩Hω2 and note that M ≺ Hω2 , M ∩ Pf = N ∩ Pf , and s ∈M . We definet = (p,M∪ {M}), observe that t ≤ s and argue that t is (N,Pf )-generic. Thegenericity of the condition t will follow from the following slightly more generalresult.

Lemma 10. Let N ≺ Hθ be countable and let t = (p,M) be a condition such thatN ∩Hω2 ∈M. Then t is (N,Pf )-generic.

Proof. Let D ∈ N be dense, so that we need to produce a condition in D ∩ Ncompatible with t. We may as well assume that t ∈ D, otherwise extend it to acondition in D and apply the argument below to the extension.

Let M = N ∩Hω2 and δ = M ∩ω1 = N ∩ω1. We notice that P∩M = P∩N andD∩M = D∩N . As above we define t ¹M = (p ¹ δ,M∩M), so that t ¹M ∈M andt ≤ t ¹ M . Let q = p ¹ (ω1 \ δ). It will suffice to produce t = (p,M) ≤ t ¹ M suchthat t ∈ D∩M and p∪q is good. For if we have such a t then easily (p∪q,M∪M)is a condition, so that t is compatible with t which lies in D ∩N .

So we suppose for a contradiction that no such condition t exists. We define Bto be the set of pairs (α, r) such that

(1) r is a finite partial function from ω1 to ω with |r| = |q|.(2) dom(p ¹ δ) < α ≤ dom(r).(3) p ¹ δ ∪ r is good.(4) There is no condition (p∗,M∗) such that

(a) (p∗,M∗) ≤ t ¹M .(b) (p∗,M∗) is below α.(c) p∗ ∪ r is good.(d) (p∗,M∗) ∈ D.

We claim that (δ, q) ∈ B. Suppose not: then we may find an extension (p∗,M∗)of t ¹M which is below δ, lies in D and is such that p∗ ∪ r is good. As we pointedout above, p∗ ∈ M but M∗ may lie outside M . However if M∗ = {M∗0 , . . .M∗k−1}then {M∗i ∩ ω1 : i < k} is in M , and using elementarity we may find a condition(p∗,M∗∗) ∈ M which lies in D with M∗∗ = {M∗∗0 , . . .M∗∗k−1} and M∗i ∩ ω1 =M∗∗i ∩ ω1 for i < k. But this is impossible by our assumption that no t as aboveexists.

Now B ∈ N because B is defined from parameters in N . Since B ∈ Hω2 , in factB ∈ M . It follows that since (δ, q) ∈ B the set {α : ∃r (α, r) ∈ B} is unbounded


in ω1, else it could be bounded by an ordinal in M . So we may construct in M asequence of elements (δi, ri) ∈ B such that δi < dom(ri) < δi+1.

Appealing to Lemma 9 we get an uncountable subsequence of this sequenceconsisting of pairs for which p∪ri is good. If we now choose such a pair with p < δithen t ≤ t ¹ M , t is below δi, p ∪ ri is good and t ∈ D; this is a contradictionbecause (δi, ri) ∈ B and the last clause in the definition of B rules this out!

¤

Now that we have showed that Pf is proper, a routine application of the PFAshows that for any (< ω)-bounded colouring of pairs from ω1, ω1 is the union of ωpolychromatic sets. This establishes Theorem 2.

In particular this shows that under PFA, ω1 →poly (ω1)22−bd. Actually we get

something a bit stronger, namely that there is a stationary polychromatic set.

Remark 4. As we mentioned in the introduction, Todorcevic proved [?] that theconclusion of Theorem 2 is consistent relative to the consistency of ZFC, and alsothat it follows from PFA. The forcing argument of that paper is rather differentfrom that of this section: it involves a mixed (finite/countable) support iterationof Cohen forcing and Jensen’s “fast club forcing”, followed by a finite support ccciteration to add the polychromatic sets. The argument from PFA involves showingthat after adding a fast club, the poset whose conditions are finite polychromaticsets separated by points of the fast club has the ccc.

Remark 5. Todorcevic [?] points out that his methods give the following ZFCresult: for every countable ordinal α, every 2-bounded colouring of pairs from [ω1]2

has a closed polychromatic set of order type α. This also follows from our results:we can add a stationary polychromatic set and then shoot a club through that setwithout collapsing ω1, then borrow an argument from [?] and observe that the treeof attempts to build a closed polychromatic set of type α must have a branch in V .Yet another proof would be to apply Galvin’s dual colouring idea as described inSection 2 to Schipperus “topological Baumgartner-Hajnal theorem” [?].

4. MA does not suffice for Theorem 2

Since the relation ω1 →poly (ω1)22−bd holds under PFA but fails under CH, it

is natural to ask about its status under Martin’s Axiom (MA). It turns out to beindependent of MA, in fact we can get the consistency of a “ccc-indestructible badpartition”.

Theorem 3. It is consistent that there is a 2-bounded normal colouring of pairsfrom ω1 which has no uncountable polychromatic set in any ccc forcing extension.

In particular since we can force MA by a ccc forcing extension over the modelof Theorem 3, MA does not imply that ω1 →poly (ω1)2

2−bd. This proof uses similartechniques to those in a paper by Abraham and Todorcevic [?] about indestructibleproperties in topology.

4.1. The poset Q. We describe the poset Q which we will ultimately use to addthe “bad colouring”. We fix for each β < ω1 a set Wβ ⊆ ω1, such that |Wβ | = ωand the sets Wβ are pairwise disjoint.

Conditions in Q are finite partial functions q such that(1) dom(q) = [F ]2 for some finite F .


(2) q(α, β) ∈Wβ .(3) q is 2-bounded.

The ordering is extension. Abusing notation a bit we refer to the finite set F asthe domain of q and write F = dom(q).

The demand that q(α, β) ∈ Wβ makes the normality of the generic colouringautomatic.

Lemma 11. For every α < ω1 the set of q with α ∈ dom(q) is dense.

Proof. Let F = dom(q). Enumerate the pairs in [F∪{α}]2\[F ]2 as (αi, βi) for i < Nand then colour them, making sure to give (αi, βi) a new colour from Wβi . ¤

The chain condition argument is also routine but we do it carefully to stress apoint about amalgamation of conditions which will be crucial later.

Lemma 12. The poset Q has the ω1-Knaster property, in particular it is ccc.

Proof. Let 〈qi : i < ω1〉 be an ω1-sequence of conditions. By a standard applicationof the ∆-system lemma we may assume that the sets Fi =def dom(qi) form a “head-tail-tail ∆-system”. That is to say there are finite subsets r and si (for i < ω1) ofω1 such that r < s0 < s1 . . . and Fi = r ∪ si.

Since W =def ∪β∈rWβ is countable and qi : [r]2 → W for all i, we may furtherassume that there is a fixed p such that qi ¹ [r]2 = p for all i. Now let i < j < ω1

and consider qi and qj . We see that [Fi]2 ∩ [Fj ]2 = [r]2 so that qi ∪ qj is a function.What is more qi ∪ qj is 2-bounded, because pairs with top points in si assumecolours in ∪β∈SiWβ while pairs with top points in sj assume colours in the disjointset ∪β∈SjWβ .

Now [Fi ∪ Fj ]2 = [Fi]2 ∪ [Fj ]2 ∪ si × sj , so to produce a common extension of qiand qj we need to assign colours to pairs in si × sj . This is possible because thereare infinitely many unused colours in Wβ for each β ∈ sj ; in particular (and this isthe point we wished to stress) we have great freedom in the choice of the colouringon si × sj . ¤

Lemma 13. The generic colouring c of [ω1]2 added by Q is normal, 2-bounded andhas no uncountable polychromatic set.

Proof. The verification that the colouring is normal and 2-bounded is easy so weconcentrate on the last claim. Let A name an uncountable set. As usual we maychoose qi and αi for i < ω1 such that

(1) αi increases with i.(2) qi ° αi ∈ A.(3) αi ∈ dom(qi).(4) The sets dom(qi) form a head-tail-tail ∆-system with root r, and qi ¹ [r]2

is independent of i.Now we choose i < j < k so that αi, αj , αk /∈ r and then (as in the proof of Lemma12) we may easily find q ≤ qi, qj , qk forcing that (αi, αk) gets the same colour as(αj , αk). ¤

4.2. Motivation for the main construction. We digress to discuss some consid-erations which motivate the combinatorics of the main construction. This digressionis purely for motivation, the impatient reader can skip everything except Definition9.


Recall that our ultimate goal is to produce a normal 2-bounded c : [ω1]2 → ω1

which has no uncountable polychromatic set in any ccc forcing extension. We aresearching for a property which will guarantee this and is consistent. Clearly ifwe are to have any chance of success then c must start off as a partition with nouncountable polychromatic set. Let us see whether this property is preserved byccc forcing.

So let us be given a normal 2-bounded c : [ω1]2 → ω1 a ccc poset R, and aR-name A for an uncountable subset of ω1. Assume that c has

Property 1: c has no uncountable polychromatic set, that is to say for every un-countable B ⊆ ω1 there exist ordinals α < β < γ in B with c(α, γ) = c(β, γ).

As usual we may produce ri and αi for i < ω1 so that the αi are increasing withi and ri ° αi ∈ A. Let B = {αi : i < ω1}. Then

(1) By our assumption on c there are many triples i < j < k with c(αi, αk) =c(αj , αk).

(2) It follows from the ccc (see Lemma 15 below) that there are many triplesi < j < k such that there is some r ≤ ri, rj , rk.

Unfortunately there is no reason to believe that there is any overlap between thesetwo classes of triples. So a natural thing to try might be to try and make the firstset of triples larger by considering

Property 2: For every uncountable B ⊆ ω1 there exists an uncountable set B′ ⊆ Bsuch that for every triple α < β < γ from B′ we have c(α, γ) = c(β, γ).

This property is preserved by ccc forcing but unfortunately it is incompatiblewith the 2-boundedness of c. We modify property 2 as follows.

Definition 9. An increasing sequence of pairs of ordinals is a sequence 〈yi : i < i∗〉where each yi is an (unordered) pair of ordinals and i < j =⇒ max yi < min yj.

Given pairs of countable ordinals a = {α, β} and b = {γ, δ} with α < β < γ < δwe say that G(a, b) holds (for c) if and only if c(α, ζ) = c(β, ζ) for some ζ ∈ b.

Now consider

Property 3: for every increasing sequence of pairs of countable ordinals 〈yi : i < ω1〉,there is an uncountable B ⊆ ω1 such that for every pair i < j from B we haveG(yi, yj).

Property 3 is preserved by ccc forcing, implies that c has no uncountable poly-chromatic set and is not obviously inconsistent with 2-boundedness. Better still, itis true in V Q that for any given sequence y = 〈yi : i < ω1〉 there is a ccc forcing Rywhich adds a witness to Property 3; the forcing is easy to describe, conditions arejust finite sets a ⊆ {yi : i < ω1} such that for any i < j, yi, yj ∈ a =⇒ G(yi, yj).

To see that Ry is ccc in V Q we verify that Q ∗ Ry is ccc in V . So let (qi, ai) fori < ω be conditions in Q ∗ Ry. As usual we may assume (extending qi and thinningout as necessary) that

(1) qi determines the value of ai, say qi ° ai = ai.(2) ai ⊆ dom(qi).(3) The sets dom(qi) form a head-tail-tail ∆-system with root r, and qi ¹ [r]2

is constant.


It is now easy to see that if i < j then as in the proof of Lemma 12 we mayfind q ≤ qi, qj such that q ° ai ∪ aj ∈ Ry. Unfortunately when we try to iteratethis forcing we run into trouble with ccc; analysing a two-step iteration Ry ∗ Rzin the same way, we may have difficulties amalgamating conditions (qi, ai, bi) and(qj , aj , bj). The amalgamation problem can be cured if each pair zi ∈ z is “widelyspaced” in a sense to be made precise in the next section. So property 3 is notquite what we want, we need to allow “spacing out” of the pairs. It is tempting totry

Property 4: for every uncountable A ⊆ ω1 there is an increasing sequence of pairs〈yi ∈ [A]2 : i < ω1〉 such that G(yi, yj) for all i < j < ω1.

However property 4 is not obviously preserved by ccc forcing. The property(∗) which we will use, and which is defined in Definition 10, is a strengthening ofproperty 4 which is easily seen to be preserved by ccc forcing. To add a witness toan instance of (∗) we can force with Ry for a well-chosen y, but need to see thateach instance of Ry which we force with is ccc.

To see that the various instances of Ry which we are forcing with are ccc we willmaintain an inductive hypothesis “there exists a club C with P (C)” where P (C)is defined in Section 4.5. P (C) implies the following statement Q(C): “if the pairsin y are each separated by a point of C then any sequence of disjoint conditionsin Ry contains two compatible conditions”. Unfortunately the statement Q(C) isnot strong enough to propagate forwards by induction through the stages of ouriteration: the statement P (C) is a sort of “n-dimensional” version of Q(C) whichcan be inductively propagated.

4.3. The property (∗).Definition 10. Let c : [ω1]2 → ω1. c has property (∗) if and only if for everyclub C ⊆ ω1, every unbounded set A ⊆ ω1 and every f : A × ω1 → A such thatf(ζ, η) ≥ η for all ζ, η there is a sequence 〈xi : i < ω1〉 such that

(1) xi = (αi, βi, γi) where αi < βi ≤ γi with αi, γi ∈ A and βi ∈ C.(2) γi = f(αi, βi)(3) If i < j < ω1 then γi < αj and G({αi, γi}, {αj , γj}) holds.

Lemma 14. If c has (∗) then c has no uncountable polychromatic set.

Proof. Suppose for a contradiction that A is an uncountable polychromatic set anddefine f(α, β) = min(A\β). Let C = ω1 and find xi as in (∗). Then for any i < j weget that αi < γi < αj < γj where all four ordinals are in A and c(αi, ζ) = c(γi, ζ)for some ζ ∈ {αj , γj}. So A is not polychromatic, contradiction! ¤

We now show that property (∗) is ccc indestructible. This uses two easy andwell known observations about ccc forcing.

Lemma 15. Let P be ccc and let 〈pi : i < ω1〉 be any sequence of conditions in P.Then some condition pi forces that {i : pi ∈ G} is uncountable.

Proof. Suppose not, and choose for every i a condition qi ≤ pi and an ordinal βisuch that qi ° {j : pj ∈ G} ⊆ βi. Then qi ° pj /∈ G for all j ≥ βi, that is to saythat qi ⊥ pj for j ≥ βi. If we now define a club set E = {j : ∀i < j βi < j} theneasily {qi : i ∈ E} is an antichain, contradicting the ccc of P. ¤


Lemma 16. Let P be ccc and let E be a name for a club subset of ω1. Then thereis a club set D such that ° D ⊆ E.

Proof. Let f name the map α 7→ min(E \ α) and let Aα = {β : ∃p p ° f(α) = β}.Then by ccc Aα is countable. If we let D = {β : ∀α < β Aα ⊆ β} then everyelement of D is forced to be a limit point of E, and hence a member of E. ¤

We use the preceding lemma to show that we only need consider club sets in Vin the next lemma.

Lemma 17. If (∗) holds then it remains true in any ccc forcing extension.

Proof. Let (∗) hold in V and let P be ccc. Let p force that C is a club subset ofω1, A is an unbounded subset of ω1, and f : A × ω1 → A is such that f(ζ, η) ≥ η

for all ζ, η. Fix D ⊆ ω1 a club set such that p ° D ⊆ C.In V let A = {α : ∃q ≤ p q ° α ∈ A}, and choose for each ζ ∈ A and η ∈ ω1 a

condition p(ζ, η) ≤ p which forces that ζ ∈ A and determines the value of f(ζ, η) assome ordinal f(ζ, η). Note that p(ζ, η) forces that f(ζ, η) ∈ A so that f(ζ, η) ∈ Aand so f : A× ω1 → A.

Now we appeal to (∗) in V to find a suitable sequence of triples xi = (αi, βi, γi)with βi ∈ D. If pi = p(αi, βi) then pi forces that αi, γi ∈ A and f(αi, βi) = γi. ByLemma 15 there is pj which forces that {i : pi ∈ G} is unbounded. Then pj ≤ pand pj forces that 〈(αi, βi, γi) : pi ∈ G〉 is a suitable witness to the truth of (∗) inV P. ¤

4.4. A combinatorial lemma. At the heart of the proof is a combinatorial lemmafrom a paper of Abraham and Todorcevic [?]. For the reader’s convenience wereproduce the proof here.

Definition 11. Let Y = 〈{αi, βi} : i < i∗〉 be an increasing sequence of pairs ofordinals.

(1) An ordinal γ is a closure point of Y if and only if ∀i < i∗ (αi < γ =⇒βi < γ). C(Y ) is the class of closure points of Y .

(2) The sequence Y is separated by A if and only if for every i < i∗ there isη ∈ A with αi < η < βi, and for every i < i + 1 < i∗ there is η ∈ A withβi < η < αi+1.

Definition 12. (1) A matrix of pairs is a finite sequence (Y1, . . . Yn) such thateach Yi is a finite increasing sequence of pairs of countable ordinals, andfor each i with 1 < i ≤ n the sequence Yi is separated by

⋂j<i C(Yj). We

say that n is the length of the matrix, and we refer to the finite sequenceYj as the jth column of the matrix.

(2) A matrix of pairs M = (Y1, . . . Yn) is separated by A if and only if thesequence Y1 is separated by A.

(3) If M = (Y1, . . . Yn) is a matrix of pairs then we say a pair of ordinalsappears in M if it is an entry in some Yj, and an ordinal appears in M ifit is a member of some pair appearing in M .

(4) If M and N are matrices of pairs then M < N if every ordinal appearingin M is less than every ordinal appearing in N . Similarly M < β if everyordinal appearing in M is less than β, and β < M if β is less than everyordinal appearing in M .


The key fact about matrices of pairs is the following lemma, which appears asLemma 5.3 in [?].

Lemma 18. Let M be a matrix of pairs. Then M has a 1-1 choice function, that isto say there is a 1-1 function which maps each pair u appearing in M to a memberof u.

Proof. We will establish the stronger claim that for every matrix of pairs M =(Y1, . . . Yn) and every ordinal ρ, there is a 1-1 choice function for M which does nottake the value ρ. The proof is by induction on n, and for fixed n by induction on|Yn|.

There is no problem for n = 1. Suppose that n > 1 and that |Yn| > 0, andchoose a pair u = {γ, δ} appearing in Yn. Since M is a matrix of pairs there is ηsuch that γ < η < δ and η ∈ C(Yi) for all i < n.

The key point is that for every pair {α, β} appearing in (Y1, . . . Yn−1), eitherα < β < η or η ≤ α. We now define new matrices by (informally) “dropping thepair {γ, δ} from column n and using η to split the earlier columns”.

Formally we define matrices of pairs M low = (Y low1 , . . . Y low

n ) and Mhigh =(Y high

1 , . . . Y highn ) as follows:

(1) For 1 ≤ i < n, the pairs appearing in Y lowi are the pairs {α, β} appearing

in Yi with α < β < η.(2) The pairs appearing in Y low

n are the pairs {α, β} appearing in Yn withβ < γ.

(3) For 1 ≤ i < n, the pairs appearing in Y highi are the pairs {α, β} appearing

in Yi with η ≤ α < β.(4) The pairs appearing in Y high

n are the pairs {α, β} appearing in Yn withα > δ.

Note that the induction hypothesis applies to both M low and Mhigh since we madesure to drop a pair from the last column Yn of the matrix M . Note also that allpairs appearing in M low consist of ordinals which are less than η, and all pairsappearing in Mhigh consist of ordinals which are greater than or equal to η.

Now suppose that the ordinal ρ which we wish to avoid has ρ < η. By theinduction hypothesis we may find a 1-1 choice function for M low avoiding the valueρ and a 1-1 choice function for Mhigh avoiding the value δ. By the remarks inthe last paragraph the ranges of these functions are disjoint, and also the union ofthese functions avoids both the values ρ and δ. To finish we choose from {γ, δ} theelement δ. Similarly if ρ ≥ η we take the union of a 1-1 choice function for M low

avoiding the value γ and a 1-1 choice function for Mhigh avoiding the value ρ, andchoose γ from {γ, δ}. ¤

4.5. The property P (C). We are almost ready to describe the main construction.We will start with V a model of GCH and force with Q. Working over V Q, we willdo a certain ccc finite support iteration Pω2 of length ω2 (whose details will be givenin the next section) to produce a model in which c has (∗), where c : [ω1]2 → ω1

is the generic colouring added by Q. A key technical point will be that each stagei < ω2, there is a club set Ci ⊆ ω1 such that a certain statement P (Ci) holds inV Pi .

Definition 13. Let C ⊆ ω1 be club and let c : [ω1]2 → ω1.


(1) Let M = (Y1, . . . Yn) and N = (Z1, . . . Zn) be matrices of pairs with thesame length n. Then G(M,N) holds if and only if M < N and for all iwith 1 ≤ i ≤ n, all pairs u appearing in Yi and all pairs v appearing in Zi,G(u, v) holds.

(2) A sequence of matrices of pairs 〈Mi : i < i∗〉 is an increasing sequence ofmatrices separated by C if and only if(a) Each Mi is separated by C.(b) For i < j there is β ∈ C such that Mi < β < Mj.

(3) P (C) holds (for C) if and only if for every 〈Mi : i < ω1〉 an increasingsequence of matrices separated by C, there exist i < j such that G(Mi,Mj).

We will inductively construct a sequence of clubs Ci so that P (Ci) holds after isteps of the iteration over V Q. The following lemma says we may take C0 = ω1.

Lemma 19. P (ω1) holds in V Q.

Proof. Let 〈Mi : i < ω1〉 be a Q-name for an increasing sequence of matrices ofpairs. As usual we may choose conditions qi and matrices of pairs Mi for i < ω1

such that(1) qi ° Mi = Mi.(2) 〈Mi : i < ω1〉 is an increasing sequence of matrices of pairs, and all the Mi

have the same length n. Let Mi = (Y 1i , . . . Y

ni ).

(3) All ordinals appearing in Mi are in dom(qi).(4) The sets dom(qi) form a head-tail-tail ∆-system with root r, and qi ¹ [r]2

is constant.(5) sup(r) < Mi.

Choose i, j with i < j and use Lemma 18 to find a 1-1 choice function F forthe pairs in Mj . Enumerate the pairs in Mj as vk for k < N , where vk appears inY dkj , that is column dk of the matrix Mj . Now for each k in turn we run throughthe pairs u appearing in column Y dkj in increasing order; if u = {ζ, η} then wewill assign the pairs {ζ, F (vk)} and {η, F (vk)} a legal colour not used before. Atthe end we extend qi ∪ qj to force these colour assignments and we have producedq ≤ qi ∪ qj such that q ° G(Mi,Mj). ¤

The argument that at every stage of the iteration there is a club set C ⊆ ω1 withP (C) is modelled on the proof that ccc is preserved in a finite support iteration.We will leave the successor step to the next section. For the limit step we need acharacterisation of how P (C) comes to hold in a ccc generic extension. Again theargument is modelled on those of [?].

Lemma 20. Let P be ccc, let p ∈ P, let C ⊆ ω1 be club and c : [ω1]2 → ω1. Thenthe following are equivalent.

(1) p °P P (C).(2) For every sequence 〈(pi,Mi) : i < ω1〉 where pi ≤ p and 〈Mi : i < ω1〉 is an

increasing sequence of matrices of pairs separated by C, there exist i < jsuch that pi is compatible with pj and G(Mi,Mj).

Proof. Suppose that p °P P (C). By Lemma 15 we may force below some pi and geta generic filter G 3 p such that pi ∈ G for unboundedly many G. Now 〈Mi : pi ∈ G〉is (relabelling) an increasing ω1-sequence of matrices of pairs which is separated byC, so we may find pi, pj ∈ G with G(Mi,Mj). Of course pi and pj are compatible


since they lie in the filter G, and the statement G(Mi,Mj) is absolute between Vand V [G].

Now suppose that 2 holds and q ≤ p. Fix 〈Mi : i < ω〉 a P-name for anan increasing ω1-sequence of matrices of pairs which is separated by C. Chooseqi ≤ q and Mi such that qi ° Mi = Mi, thinning out if necessary we may assumethat 〈Mi : i < ω1〉 is an increasing sequence of matrices of pairs separated by C.Appealing to 2 there are i < j so that qi, qj are compatible and G(Mi,Mj), so thatif r ≤ qi, qj then r ° G(Mi, Mj). We are done by the usual density argument. ¤Lemma 21. Let δ < ω2 be a limit ordinal, and let 〈Ci : i < δ〉 be a sequence ofclub subsets of ω1.

If Pδ is a ccc finite support iteration of limit length δ < ω2, such that for everyi < δ the statement P (Ci) holds in V Pi , then there is Cδ ⊆ ω1 club such that P (Cδ)holds in V Pδ .

Proof. There are two cases.

cf(δ) = ω. Choose 〈δn : n < ω〉 cofinal in δ and set Cδ =⋂n Cδn .

We verify the condition from Lemma 20. Let 〈(pi,Mi) : i < ω1〉 where pi ∈ Pδand 〈Mi : i < ω1〉 is an increasing sequence of matrices of pairs separated by Cδ.Since the iteration is done with finite support, for every i < ω1 there is n < ω suchthat supp(pi) ⊆ δn. Thinning out if necessary we may assume that there is a fixedn such that supp(pi) ⊆ δn for all i.

By the choice of Cδ, we have Cδ ⊆ Cδn . So we have a sequence 〈(pi,Mi) : i < ω1〉where (morally speaking) pi ∈ Pδn and 〈Mi : i < ω1〉 is an increasing sequence ofmatrices of pairs separated by Cδn .

Since P (Cn) holds in V Pδn , another appeal to Lemma 20 shows that there arei < j so that pi, pj are compatible and (Mi,Mj). We are done.

cf(δ) = ω1. Choose 〈δj : j < ω1〉 cofinal in δ and let Cδ be the diagonal intersectionof the Cδj ; by construction for each j we have that Cδ \ Cδj is bounded.

Again we verify the condition from Lemma 20. Let 〈(pi,Mi) : i < ω1〉 wherepi ∈ Pδ and 〈Mi : i < ω1〉 is an increasing sequence of matrices of pairs separated byCδ. Appealing to the ∆-system lemma we may assume, thinning out if necessary,that the finite sets supp(pi) form a ∆-system with some root r. We fix i∗ suchthat r ⊆ δi∗ ; note that if i < j then pi, pj are compatible in Pδ if and only if therestrictions pi ¹ δi∗ , pj ¹ δi∗ are compatible in Pδ∗ .

The sequence 〈Mi : i < ω1〉 is separated by Cδ and Cδ \ Cδi∗ is bounded. Sothrowing away an initial segment if necessary, we may assume that 〈Mi : i < ω1〉 isseparated by Cδi∗ . Recalling that P (Cδi∗ ) holds in V Pδ∗ and applying Lemma 20to the sequence 〈(pi ¹ δi∗ ,Mi) : i < ω1〉 we obtain i < j such that pi ¹ δi∗ , pj ¹ δi∗are compatible and G(Mi,Mj). By the remarks in the last paragraph pi and pj arecompatible so we are done. ¤4.6. Putting it all together. Recall that the plan of the proof is as follows: startwith GCH in V , force with Q to add a colouring c, and then build a ccc iterationPω2 to force that c has (∗), constructing along the way a sequence of clubs Ci sothat P (Ci) holds in V Q∗Pi .

The next lemma is the final piece in the puzzle; it shows that we can go one stepin the construction, that is assuming P (D) we may force an instance of (∗) in a cccfashion so that in the extension P (E) holds for some E.


Lemma 22. Let c : [ω1]2 → ω1 and let property P (D) hold for some club setD ⊆ ω1. Let f,A,C be as in the hypotheses of (∗), that is

(1) A is an unbounded subset of ω1.(2) f : A× ω1 → A and f(ζ, η) ≥ η for all ζ, η.(3) C is a club subset of ω1.

Then there are a ccc poset R and a club set E ⊆ ω1 such that in the extensionby R

(1) The conclusion of (∗) holds; that is there is a sequence 〈xi : i < ω1〉 suchthat(a) xi = (αi, βi, γi) where αi < βi ≤ γi with αi, γi ∈ A and βi ∈ C.(b) γi = f(αi, βi)(c) If i < j < ω1 then γi < αj and G({αi, γi}, {αj , γj}) holds.

(2) P (E) holds.

Proof. We start by fixing a sequence 〈yi : i < ω1〉 such that(1) yi = (αi, βi, γi) where αi < βi ≤ γi with αi, γi ∈ A and βi ∈ C ∩D.(2) γi = f(αi, βi)(3) If i < j < ω1 then there is η ∈ C ∩D such that γi < αj .

Let z = 〈zi : i < ω1〉 where zi = (αi, γi), and note that z is an increasingsequence of pairs separated by D. To produce a witness for (∗) we need to producea subsequence such that G holds for each pair from the subsequence; this is preciselywhat R will do.

We define R to be the set of finite sets a ⊆ {zi : i < ω1} such that for all i < j,zi, zj ∈ a =⇒ G(zi, zj). The ordering is by inclusion.Claim 1: R is ccc.Proof of Claim 1: Let 〈ai : i < ω1〉 be conditions in R. As usual we may assumethat the sets

⋃ai form a head-tail-tail ∆-system with some root r. Accordingly we

write ai = r ∪ bi where each bi may be regarded as a finite increasing sequence ofpairs.

Now the sequence 〈bi : i < ω1〉 may be regarded as an increasing sequence ofmatrices of pairs (each of length 1) and what is more it is separated by D. Appealingto the property P (D) in the n = 1 case we get i < j so that G(bi, bj), from whichit follows easily that ai ∪ aj is a condition.Claim 2: If E = C(z) ∩D then E is a club subset of ω1 and P (E) holds in V R.Proof of Claim 2: It is routine to check that E is club. To show that P (E) holds inV R, once more we appeal to Lemma 20. Let 〈(ri,Mi) : i < ω1〉 be such that ri ∈ Rand 〈Mi : i < ω1〉 is an increasing sequence of matrices of pairs separated by E.

By a ∆-system argument as in the proof that R is ccc we may assume thatri = r ∪ si, where 〈si : i < ω1〉 is an increasing sequence of matrices of pairs eachof length 1.

Then easily if we define matrices Ni so that Ni,1 = si and Ni,j+1 = Mi,j for 1 ≤j ≤ lh(Mi), the sequence 〈Ni : i < ω1〉 is an increasing sequence of matrices of pairsseparated by D. Appealing to P (D) in V we get i < j such that G(Ni, Nj), whichamounts to saying that G(si, sj) (so that ri, rj are compatible) and G(Mi,Mj). ByLemma 20 we showed that P (E) holds in V R.

¤The proof of Theorem 3 is now completely routine. We force with Q and then

build a ccc iteration Pω2 to force (∗), along with a sequence in V of clubs Ci such


that P (Ci) holds in V Q∗Pi . At each stage i we will use Lemma 22 to handle someinstance of (∗) corresponding to fi, Ai, Ci, the Lemma only guarantees that there isCi+1 ∈ V Pi which works but we may appeal to Lemma 16 to make a suitable choiceof Ci+1 in V . The same book-keeping arguments that are used in the consistencyproof for Martin’s Axiom show that in ω2 steps we can handle all possible f,A,Cso that (∗) holds in V Q∗Pω2 . Working over this model we can now force MA (or anyother statement which can be forced by ccc forcing) while preserving (∗).

After we proved the results of this section we learned of some related work [?] bySoukup, also using similar methods to those of [?]. Soukup builds a very generalframework for making combinatorial properties of a certain kind indestructibleunder ccc forcing; his intended applications are topological, but a proof of Theorem3 can easily be read off from his results.

More specifically given a set K, we consider a graph G with vertex set ω1 ×K;the graph is said to be m-solid if and only if for every ω1-sequence of finite partialfunctions from ω1 to K where the domains are pairwise disjoint sets of size m, thereexist members s and t of the sequence such that s× t is contained in the edge setof G. G is strongly solid if and only if G is m-solid for all m. Soukup shows thatif 2ω1 = ω2, |K| ≤ ω2 and G is strongly solid, then for each m there is a ccc posetof size ω2 which makes the m-solidity of G become ccc-indestructible.

Soukup uses this result to get certain topological statements consistent with MA.Among these statements are “there is an uncountable first countable space in whichevery open set is countable or co-countable”, “there is an S-group”, and “there isan L-group”. In rough outline the general idea is to construct a subspace X of2ω1 and an associated strongly solid graph GX such that the desired property of Xcorresponds to the m-solidity of GX for some m.

5. A ccc destructible partition

In the light of the results from the preceding section it is natural to ask for aconcrete example of a 2-bounded normal colouring of [ω1]2 which has no uncount-able polychromatic set in V , but acquires one in some ccc generic extension. It isroutine to see that a poset with the Knaster property cannot add an uncountablepolychromatic set for a colouring with no such set, so the obvious forcing poset totry is an (inverted) Souslin tree. This turns out to work.

Theorem 4. Let T be a Souslin tree and let CH hold. Then there is a colouringf : [T ]2 → ω1 such that

(1) f is 2-bounded and has no uncountable polychromatic set.(2) For every L ⊆ T , if L is linearly ordered by <T then L is polychromatic.

In particular an uncountable polychromatic set is added by the ccc forcing PT =(T,≥T ), which adds an uncountable branch in T .

Proof. Fix a Souslin tree T , where without loss of generality we may assume thatthe underlying set of the tree is ω1. We adopt the conventions that

Tα = {t ∈ T : ht(t) < α},and

Levα = {t ∈ T : ht(t) = α}.We may assume without loss of generality that T is normal. That is to say

(1) T has a unique element on level zero.


(2) Every element has at least two immediate successors.(3) Every point on every limit level is determined uniquely by its set of prede-

cessors.(4) For all α, β with α < β < ω1 and all s ∈ Levα there is t ∈ Levβ with

s <T t.

The following property of some subsets of T will be helpful in the construction: saythat X ⊆ T is far from linear if and only if X is not the union of a linearly orderedset and a finite set.

Fix an enumeration of the countable subsets of ω1 as Xi for i < ω1. Fix adecomposition of ω1 into pairwise disjoint countably infinite sets Ai for i < ω1. Fixalso, for each β < ω1, a decomposition of A2β+1 as the union of infinite disjointsets A0

2β+1 and A12β+1. We will build a function f : [T ]2 → ω1 such that

(1) For all distinct s, t ∈ T if α = ht(s) = ht(t) then f({s, t}) ∈ A2α. Ifht(s) < ht(t) then f({s, t}) ∈ A2t+1.1

(2) For every α < ω1, f ¹ [Levα]2 is 1-1.(3) For all s, t, u ∈ T , if s <T t <T u then f({s, u}) 6= f({t, u}).(4) f is 2-bounded.

We will construct f level by level. More explicitly, given f ¹ [Tα]2 we willdescribe how to extend it to f ¹ [Tα+1]2. Note that pairs in [Tα+1]2 \ [Tα]2 areof two kinds, either they are drawn from [Levα]2 or they of the form {s, t} whereht(s) < ht(t) = α.

Suppose that we are given f ¹ [Tα]2. We define f ¹ [Levα]2 to be any 1-1 mapfrom [Levα]2 to A2α.

To colour the pairs {s, t} where ht(s) < ht(t) = α, we proceed as follows. Wefirst enumerate the set of those Xi with i < α such that

(1) Xi ⊆ Tα,(2) Xi is far from linear,(3) Xi is polychromatic for f ¹ [Tα]2,

as 〈Yj : j < ω〉. We choose inductively αj , βj ∈ Yj for j < ω such that

(1) αj and βj are not comparable in T .(2) αj , βj /∈

⋃j′<j{αj′ , βj′}

This is possible because each Yj is far from linear.Now for each element t of Levα we proceed as follows. We set f({αj , t}) =

f({βj , t}) = ctj where the ctj are pairwise distinct elements of A02t+1. We then

enumerate the set Tα \⋃j<ω{αj , βj} as γk for k < ω, and set f({γk, t}) = dtk

where the dtk are pairwise distinct elements of A12t+1. For use a bit later we note

that if s, t are distinct elements of Tα, u ∈ Levα and f({s, u}) = f({t, u}), then{s, t} = {αj , βj} for some j.

We should check that the various properties of f are preserved. Clauses 1 and 2are immediate because we chose colours from the appropriate sets, and made surethat f was 1-1 on the set of pairs from the new level Levα.

We claim that we have preserved clause 3. We know that f ¹ [Tα]2 satisfiesclause 3 so suppose for a contradiction that s <T t <T u with ht(u) = α andf({s, u}) = f({t, u}). As we pointed out above, this implies that there is some

1Remember that t is itself a countable ordinal.


j with {s, t} = {αj , βj}. This is impossible because αj , βj were chosen to beincomparable.

For clause 4 we analyse how two pairs from Tα+1 may come to have the samecolour. Suppose that f({s, u}) = f({t, v}) = c for s, t, u, v ∈ Tα+1, where {s, u} 6={t, v}. Suppose that c ∈ Ai. If i is even then all of s, t, u, v must be on the samelevel, but that is impossible by clause 2. So i is odd, say i = 2j + 1. By clause 1we see that without loss of generality ht(s) < ht(u) and u = j, and ht(t) < ht(v)and v = j. Summarising the analysis: if f({s, u}) = f({t, v}) for s, t, u, v ∈ Tα+1

then u = v and ht(s), ht(t) < ht(u).We claim that we have preserved clause 4. Suppose for a contradiction that

{s, v}, {t, w} and {u, x} are three pairs in [Tα+1]2 such that f({s, v}) = f({t, w}) =f({u, x}). Then by the analysis from the last paragraph v = w = x, and also wehave ht(s), ht(t), ht(u) < ht(v). What is more f ¹ [Tα]2 satisfies clause 4, so thatnecessarily ht(x) = α. But then as in the discussion of 3 all three of s, t, u must beamong {αj , βj} for some j, which is impossible. This concludes the proof that wecan construct f .

By construction f is 2-bounded, and by clauses 1 and 3 any linearly orderedset is polychromatic.. So to finish we need only show that there is no uncountablepolychromatic set. Suppose for a contradiction that X is such a set. Then Xis far from linear since T is a Souslin tree, and in particular has no uncountablebranches; it is easy to see that there exists α < ω1 such that X ∩ Tα is far fromlinear. We fix i such that X ∩ Tα = Xi. Now we find u ∈ X such that u ∈ Levβfor β > α, i. The construction of f guarantees that there are distinct s, t ∈ Xi withf({s, u}) = f({t, u}), a contradiction as X is polychromatic. ¤

6. A result from Martin’s Maximum

In this section we consider 2-bounded colourings of [ω2]2. Our main result willuse the powerful forcing axiom Martin’s Maximum (MM) [?]. We say that a forcingposet P is stationary preserving if and only if P forces that every stationary subsetof ω1 remains stationary.

Martin’s Maximum (MM) is the assertion that for every stationary preserving Pand every family F of dense subsets of P with |F| = ω1, there is a filter F on Psuch that F ∩D 6= ∅ for all D ∈ F .

Theorem 5. Assuming MM, every normal 2-bounded colouring of [ω2]2 has a closedpolychromatic set of order type ω1.

The main point is to show that if c is such a colouring then we can add a closedpolychromatic set of order type ω1 with a stationary preserving forcing poset.

6.1. The poset Q(d,E). We begin by describing a poset which aims, given a 2-bounded normal d : [ω1]2 → ω1, to add a closed unbounded polychromatic subset ofω1. This poset will in general collapse ω1, it will be used as a component of our finalconstruction. In some sense our poset Q(d,E) is a descendant of Baumgartner’sposet [?] for adding a club subset with finite conditions but it also has somethingin common with the poset Pf from Section 3.

Let E be a fixed club set of countable elementary substructures of Hω2 ; fortechnical reasons we will define our poset so that only models from E may appearas side conditions. The conditions in Q(d,E) have the form p = (Mp,Bp) where


(1) Mp is a finite set {M0, . . .Mn−1} of elements of E, such that M0 ∈M1 . . . ∈Mn−1. We will denote by Xp the set {Mi ∩ ω1 : i < n}.

(2) d ∈M0.(3) Bp is a set of half-open intervals {(α0, β0], . . . (αm−1, βm−1]} where α0 <

β0 < α1 < . . . αm−1 < βm−1 < ω1.(4) The set Xp is polychromatic for d.(5) The set Xp is disjoint from

⋃Bp.Conditions are ordered as follows: p ≤ q if and only if Mq ⊆Mp and Bq ⊆ Bp.

The motivation is that p forces the elements of Xp into the generic club set andblocks out the elements of

⋃Bp. Accordingly if G is Q(d,E)-generic we defineCG =

⋃p∈GXp. Clearly CG is polychromatic, but we should verify that it is closed

and unbounded in ω1; since Q(d,E) may collapse ω1 it is unclear at this point whatwill be the order type of CG.

The following lemma will show that CG is unbounded.

Lemma 23. Let p ∈ Q(d,E) and let γ < ω1. Then there exists q ≤ p withγ < max(Xq).

Proof. Let ζ = maxXp and let M ∈ Mp be the model with M ∩ ω1 = ζ. Noticethat Xp ∩ ζ ∈M . Let

S = {η ∈ ω1 : (Xp ∩ ζ) ∪ {η} is polychromatic }Then since d ∈M we see that S ∈M , and so since ζ = M ∩ ω1 ∈ S we have by

Lemma 2 that S is stationary.Now we recall to the reader’s attention the function F from Section 3, and in

particular Lemma 3 on adding points to polychromatic sets. By elementarity wemay find T ∈ M stationary with T ⊆ S and ρ ∈ M such that for all η ∈ T ,F ((Xp ∩ ζ) ∪ {η}) ⊆ ρ. Now of course ρ ∈M ∩ ω1 = ζ, so for all η ∈ T with η > ζwe can appeal to Lemma 3 to see that Xp ∪ {η} is polychromatic.

Finally we can build a continuous increasing chain 〈Nj : j < ω1〉 with Nj ∈ E,p ∈ N0 and Nj ∈ Nj+1 for all j. The set {Nj ∩ ω1 : j < ω1} is club so we can findj so that η =def Nj ∩ ω1 ∈ T , η > γ, and η is above the maximum point of the lastinterval in Bp. It is now easy to see that q =def (Mp∪{Nj},Bp) is as required. ¤

The following lemma will show that CG is closed.

Lemma 24. Let p ∈ Q(d,E), and let γ < ω1 be a limit ordinal such that p ° γ /∈CG. Then there is q ≤ p such that q ° sup(CG ∩ γ) < γ.

Proof. Since p ° γ /∈ CG, surely γ /∈ Xp. If γ ∈ ⋃Bp we are done, so we assumethis is not the case. Since γ is limit we see that Xp ∩ γ and

⋃Bp ∩ γ are bothbounded in γ, say by δ. Now it is easy to see that q =def (Mp,Bp ∪ {(δ, γ]}) is asrequired. ¤

It will not in general be the case that Q(d,E) is proper. To see this supposethat N ≺ Hθ for some large regular θ, p ∈ Q(d,E) ∩N and Xp ∪ {N ∩ ω1} is notpolychromatic. Then if we could find an N -generic q ≤ p, by Lemma 2 q wouldforce that N ∩ ω1 is in CG; but also q would force Xp ⊆ CG, a contradiction sinceCG is polychromatic.

This discussion gives us the clue about how to find generic conditions in Q(d,E).Before we get down to the details we introduce some convenient notation similarto that we used for Pf in Section 3.


(1) If p = (M,B) is a condition in Q(d,E) and M ∈M then we define

p ¹M = (M∩M,B ∩M)

Routinely we see that p ¹M is a condition, p ¹M ∈M and p ≤ p ¹M .(2) Given a countable ordinal γ, say that a condition p is below γ if and only

if Xp ⊆ γ and⋃Bp ⊆ γ. It is true that p ¹ M is below M ∩ ω1, but false

in general that every condition below M ∩ ω1 is in M .Now we can argue that under some circumstances generic conditions do exist in

Q(d,E). The proof is similar in some respects to that of Lemma 10.

Lemma 25. Let θ be a large regular cardinal. Let N ≺ Hθ be countable withQ(d,E) ∈ N . If p ∈ Q(d,E) is such that N ∩Hω2 ∈Mp then p is N -generic.

Proof. Let M = N ∩ Hω2 . Since Q(d,E) ⊆ Hω2 , N ∩ Q(d,E) = M ∩ Q(d,E).Given a dense set D ∈ N we need to show that D ∩N is predense below p. SinceM ∈Mq for any q ≤ p, we may as well assume that p ∈ D.

Let δ = M ∩ ω1 and let q = p ¹ M . We claim that it will suffice to produce rsuch that

(1) r is below δ.(2) r ≤ q.(3) r ∈ D.(4) Xr ∪Xp is polychromatic.

For although such an r may not be in M the finite set Xr will be in M , and so byelementarity of N we will be able to find r∗ ∈ M with r∗ ≤ q, r∗ ∈ D, Xr∗ = Xr.Then easily (Mr∗ ∪ Mp,Br∗ ∪ Bp) is a condition extending both r∗ and p. Wesuppose for a contradiction that there is no r with the four properties listed above.

Now let Xp \M = {α0, . . . αt−1} where αi is increasing with i. Of course α0 =M ∩ ω1 = δ. We define a set S ∈ M of ordinals “like δ”. Specifically we let S bethe set of those β such that

(1) q is below β.(2) There exist β1, . . . βt−1 such that

(a) β < β1 < . . . βt−1 < ω1

(b) Xq ∪ {β, β1, . . . βt−1} is polychromatic.(c) There is no r such that

(i) r is below β.(ii) r ≤ q.

(iii) r ∈ D.(iv) Xr ∪ {β, β1, . . . βt−1} is polychromatic.

S ∈M because S is defined from parameters in N and lies in Hω2 , and since δ ∈ S(using βi = αi for 0 < i < t as witnesses) we see that S is stationary.

Now we enumerate the models in Mp \M as M0, . . .Mt−1 where Mi ∩ ω1 = αi,so in particular M0 = M . Now we proceed exactly as in the proof of Lemma 23 tochoose stationary sets S0, . . . St−1 such that Si ∈ Mi, S0 ⊆ S, Si+1 ⊆ Si and forevery large enough β ∈ Si there exist β1, . . . βt−1 such that

(1) β < β1 < . . . βt−1 < ω1.(2) Xq ∪ {α0, . . . αi} ∪ {β, β1, . . . βt−1} is polychromatic.(3) There is no r such that

(a) r is below β.(b) r ≤ q.


(c) r ∈ D.(d) Xr ∪ {β1, . . . βt−1} is polychromatic.

At the end of the construction choose β ∈ St−1 with p below β and suitable βias above. This is a contradiction because

Xq ∪ {α0, . . . αt−1} ∪ {β, β1, . . . βt−1} = Xp ∪ {β, β1, . . . βt−1}and is polychromatic, also p is below β, p ≤ q, p ∈ D. ¤6.2. The poset P. We define a poset P to add an increasing, continuous and cofinalmap from ω1 to ω2, preserving ω1. The poset P is just a variation on the standardcountably closed Levy collapse Coll(ω1, ω2). We will assume that P is defined in auniverse V where 2ω = ω2.

Conditions in P are functions f such that(1) dom(f) is a successor ordinal less than ω1.(2) rge(f) is a subset of ω2 ∩ cof(ω).(3) f is increasing and continuous.

The ordering is extension. It is easy to see that P is countably closed and |P| = ω2.It follows that ωV1 is preserved, ωV2 is an ordinal of cardinality and cofinality ω1 inV [G], and ωV3 = ω

V [G]2 .

To avoid confusion in the discussion that follows we let κ = ωV3 . We note thateasily HV [G]

κ = Hκ[G], because every bounded subset of κ in V [G] has a canonicalname in Hκ. We need some analysis of the countable elementary submodels ofHκ[G] in V [G].

We start by noticing that if F ∈ V is a Skolem function2 for Hκ, then in V [G],the set of countable Y ⊆ Hκ[G] with Y ∩Hκ closed under F is club. So in V [G]the set of countable N ≺ Hκ[G] such that N ∩HV

κ ≺ HVκ forms a club set E.

Now let N ∈ E and let M = N ∩ Hκ, so that M ≺ Hκ. Since P is countablyclosed, M ∈ V . What is more every element of N is coded by a bounded subset ofκ in N , and every such set has a canonical name in N ∩Hκ (that is in M), so infact N = M [G].

It is profitable to discuss this in terms of forcing and generic conditions. Let pforce that N ∈ E. Then arguing as above we can find q ≤ p such that q ° N∩Hκ =M . Automatically we have that q ° N = M [G], in particular q ° M [G] ∩ V = Mand so q is (M,P)-generic.

Note: We are not proving that M [G] ∩ V = M for all M and G. If we fixM and apply the analysis of the last paragraph to the name M [G] all we get isM [G] ∩ V = M∗ and M [G] = M∗[G] for some M∗ ⊇M , and G need only containan (M∗,P)-generic condition.

The following lemma will be useful in the main construction.

Lemma 26. Let M ≺ Hκ be countable, let δ = M ∩ ω1 and let f ∈ P be (M,P)-generic. Then δ + 1 ⊆ dom(f) and f(δ) = sup(M ∩ ω2). Also f forces thatsup(M [G] ∩ ω2) = f(δ).

Proof. Let α < δ. Then in particular α ∈ dom(f), since otherwise we can extendf to force a bad value at α+ 1. It is then clear that f(α) ∈M ∩ ω2.

Similarly let β ∈ M ∩ ω2 and consider a name for the least α < ω1 such thatthe generic function maps α to a point greater than β. This name is in M so f

2That is to say, if X ⊆ Hκ is closed under F then X ≺ Hκ.


must force that the least such α is in M , which implies there is α < δ such thatf(α) > β.

So f ¹ δ maps δ cofinally into M ∩ ω2, and since f must be continuous withdomain a successor ordinal we see that δ ∈ dom(f) and f(δ) = sup(M ∩ ω2). Thesecond claim is immediate because f forces that M [G] ∩ V = M ∩ V . ¤

6.3. Preserving stationary sets. We recall that our goal is to use MM to showthat every normal 2-bounded colouring c of [ω2]2 has a closed polychromatic set oforder type ω1. We will assume that all elementary substructures of Hω3 discussedin the proof contain c.

We will argue that a certain two-step iteration is stationary preserving. The firststep is to force with the poset P from Section 6.2. This poset is countably closedand adds a continuous increasing cofinal map g : ωV1 → ωV2 . We use g to definea colouring d ∈ V [g] by the formula d(α, β) = c(g(α), g(β)) so that easily d is anormal 2-bounded colouring of [ω1]2 The second step of the iteration is to forceover V [g] with Q(d,E) as defined in Section 6.1, where E is chosen in V [g] to bethe club set of countable N ≺ Hκ[g] such that N ∩HV

κ ≺ HVκ .

It is important to note that ωV [g]2 = ωV3 so that Q is defined using countable

elementary substructures of Hω3 [G]. To minimise confusion we carry over from thelast section the convention that κ = ωV3 .

Lemma 27. P ∗ Q(d,E) is stationary preserving.

Proof. Let us say that a condition (f, N , B) in P ∗ Q(d,E) is nice if and only if

(1) f decides the value of B.(2) There is a finite set {M0, . . .Mn−1} of countable elementary substructures

of Hκ such that(a) M0 ∩ ω1 < M1 ∩ ω1 < . . .Mn−1 ∩ ω1.(b) f ° N = {Mi[G] : i < n}.(c) f is (Mi,P)-generic for all i < n.

Note that by the discussion from the last section the set of nice conditions is dense,so we may assume that all conditions are nice.

Now let S ⊆ ω1 be stationary and let C be a P∗Q(d,E)-name for a club subset ofω1. Let p = (f, {Mi[G] : i < n},B) be a nice condition. Let δi = Mi ∩ω1 for i < n,and recall from the last section that δn−1 + 1 ⊆ dom(f) and f(δi) = sup(Mi ∩ ω2).Let ηi = f(δi), X = {δi : i < n}, Y = {ηi : i < n}.

By the definition of d and Q(d,E), we see that Y is polychromatic for c. Alsowe see that X ∩ δn−1 ∈ Mn−1 and so easily f forces that Y ∩ ηn−1 ∈ Mn−1[g], sothat in fact Y ∩ ηn−1 ∈ Mn−1. Let T be the set of ordinals η ∈ ω2 ∩ cof(ω) suchthat (Y ∩ δn−1) ∪ {η} is polychromatic for c, then T ∈Mn−1 and δn−1 ∈ T so (byan easy variation on the argument of Lemma 2) T is stationary in ω2.

Fix a large regular cardinal θ. We may find (using a standard result from theMartin’s maximum paper [?]) a countable N ≺ Hθ such that c, C, S,P ∈ N , α =def

N ∩ ω1 ∈ S, and η =def sup(N ∩ ω2) ∈ T . Let M = N ∩Hκ, so that M ≺ Hκ, andnote that f ∈M and Mi ∈M for all i < n.

Enumerating the dense subsets of P which lie in N in order type ω, and buildinga decreasing chain of elements of P ∩M to meet them all, we may readily buildf∗ ≤ f such that f is (N,P)-generic. By the discussion from the last section, wesee that α+ 1 ⊆ dom(f∗) and f∗(α) = η.


It is now routine to check that q = (f∗, {Mi[G] : i < n} ∪ {M [G]},B) is a nicecondition. What is more f∗ is (N,P)-generic and f∗ ° N [G]∩Hκ[G] = M [G], so byLemma 25 f∗ forces that ({Mi[G] : i < n}∪ {M [G]},B) is (N [G], Q(d,E))-generic.It follows that q is P ∗ Q(d,E)-generic, and so q ° β ∈ C. ¤

Once we know that P ∗ Q(d,E) is stationary preserving, a routine applicationof MM concludes the proof of Theorem 5. We note that actually the same proofwould work for (< ω)-bounded colourings.

7. PFA does not suffice for Theorem 5

We showed in Section 6 that under MM, every 2-bounded colouring of [ω2]2 hasa closed polychromatic set of order type ω1. It follows immediately from Theorem2 that under PFA every such colouring has a polychromatic set of order type ω1

which is stationary in its supremum. So it is natural to ask whether PFA would havesufficed to show that every 2-bounded colouring of [ω2]2 has a closed polychromaticset of order type ω1. We will show that this is not the case.

We consider the following combinatorial principle:

Principle W : there exists a function g such that(1) The domain of g is ω2 ∩ cof(ω).(2) For all γ ∈ ω2∩cof(ω), g(γ) is a pair of ordinals (α, β) such that α < β < γ.(3) For every δ ∈ ω2 ∩ cof(ω1) and every pair (α, β) such that α < β < δ, the

set {γ ∈ δ ∩ cof(ω) : g(γ) = (α, β)} is stationary in δ.

The motivation for introducing principle W is of course to use it to build wildcolourings.

Lemma 28. If principle W holds then there is a 2-bounded colouring of [ω2]2 withno polychromatic set which is a closed copy of ω1.

Proof. We build the colouring by induction in such a way that if g(γ) = (α, β) thenthe pairs {α, γ} and {β, γ} get the same colour. If c is a closed set of order type ω1

and α, β are the first two points of c then by property 3 above there is γ ∈ c withg(γ) = (α, β). By construction c is not polychromatic. ¤

A few remarks are in order about the principle W .(1) It would surely be equivalent to assert the existence of a regressive function

G defined on ω2 ∩ cof(ω), such that G assumes every value less than δstationarily often in every δ ∈ ω2 ∩ cof(ω). Principle W is formulated as itis for use in the proof of Lemma 28.

(2) The principle W evolved in the following way. We first constructed a colour-ing as in Lemma 28 assuming ¤ω1 , which is inconsistent with PFA (forexample from ¤ω1 we may build a special ω2-tree, while PFA implies thereare no ω2-Aronszajn trees). We then began to look for a consequence of¤ω1 strong enough to give the conclusion of Lemma 28, yet weak enoughto be consistent with PFA.

(3) If g is a witness to principle W then actually for every α < β < ω2 theset {γ ∈ ω2 ∩ cof(ω) : g(γ) = (α, β)} is stationary in ω2. To see this letC be club in ω2, let δ ∈ lim(C) ∩ cof(ω1), and use property 3 to see thatg(γ) = (α, β) for some γ ∈ C ∩ δ.


(4) Principle W is consistently false, for example it fails under MM. This followsfrom Lemma 28 and Theorem 5, but we may also give a direct proof. A well-known consequence of MM is that every stationary subset of ω2 ∩ cof(ω)contains a closed copy of ω1, but this contradicts property 3.

We describe a forcing poset P to add g witnessing principle W . We add g byforcing with initial segments. Explicitly conditions are functions p such that

(1) The domain of p is a proper initial segment of ω2 ∩ cof(ω).(2) Conditions 2 and 3 are satisfied for all γ ∈ dom(p) and all δ ≤ sup(dom(p))

with cf(δ) = ω1.

The ordering is extension. We will ultimately prove if PFA holds in V , the PFAstill holds after forcing with P.

It is clear that P is countably closed. To show that P adds no ω1-sequences ofordinals we use the idea of strategic closure. For more about strategic closure werefer the reader to Foreman’s paper about games on Boolean algebras [?].

A forcing poset P is said to be η-strategically closed for some ordinal η if andonly if player II has a winning strategy in the following game of perfect information.Players I and II collaborate to build a (non-strictly) decreasing chain in P with Iplaying at odd stages and II at nonzero even stages (including all limit stages).Play continues for at most η steps, the first player who does not move at their turnloses, and II wins if and only if play continues for η steps.

We will show that P is (ω1 + 1)-strategically closed, and implicit in the proofwill be the argument that any condition can be extended to any prescribed length.In particular forcing with P adds no subsets of ω1, so that stationary subsets of ω1

are preserved, as is the cardinal ω2. It follows that the generic function added byP is a witness to principle W .

We fix in V a decomposition of ω1 as the disjoint union of a family of stationarysets {Ti : i < ω1}. Let pi be the move at stage i of the strategic closure game andlet dom(pi) = λi ∩ cof(ω). As play proceeds II builds an enumeration of all thepairs (α, β) with α < β < supi<ω1

λi as {(αj , βj) : j < ω1}. For every limit ordinalγ ≤ ω1 she forms pγ as follows:

(1) λγ = µ+ 1 where µ = supi<γ λi.(2) pγ ¹ µ =

⋃i<γ pi.

(3) pγ(µ) = (αj , βj) for the unique j < ω1 with γ ∈ Tj , IF (αj , βj) has alreadybeen defined (so that in particular αj < βj < µ).

Clearly this is a winning strategy.We now argue that principle W is consistent with PFA. The argument is similar

to the proof by Beaudoin [?] that PFA is consistent with the existence of a non-reflecting stationary set in ω2, and also owes something to the coding argumentsfrom the Martin’s Maximum paper [?].

The obvious strategy for proving that PFA is consistent with W would be tostart with a model of PFA, force with P as above and then try to argue that PFAis preserved as follows. Given Q a P-name for a proper poset and Di for i < ω1

which are P-names for dense subsets of Q, apply the PFA in V to the proper posetP ∗ Q and the dense sets Ei = {(p, q) : p ° q ∈ Di} to get a filter F on P ∗ Q whichmeets the Ei. Then take a lower bound p for {p : ∃q (p, q) ∈ F}, so that p forcesthat {qG : ∃p (p, q) ∈ F} generates a suitable filter.


There is an equally obvious problem with this strategy, namely that no suchlower bound p may exist. The idea of the proof, which we owe to Beaudoin [?], isto define an auxiliary forcing R so that P∗ Q∗ R is proper, and that when we applyPFA to get a filter on P ∗ Q ∗ R the “R-part of F” will let us find a lower bound forthe conditions in the “P-part of F”.

We will fix in V a regressive function r on ω1 such that {α : r(α) = β} isstationary for all β < ω1. Let Q ∈ V P be a name for a proper forcing poset andlet Di for i < ω1 be P-names for dense open sets in Q. We describe an auxiliaryforcing R in V P∗Q. R is a two step iteration R0 ∗ R1. R0 is a version of the standardLevy collapse Coll(ω1, ω

V2 ), explicitly conditions are functions p with dom(p) ∈ ω1

and rge(p) ⊆ [ωV2 ]2, ordered by extension.We stress that ωV2 is the second uncountable cardinal as defined in V , which may

be collapsed when we force with Q. There is no need to write ωV1 since the iterationof proper forcing posets is again proper and hence forcing with P ∗ Q preserves ω1.

Conditions in R1 are sets d such that

(1) d is countable.(2) d is a closed and bounded subset of ω2, in particular d has a largest element.(3) Every element of d has cofinality ω.(4) For all α ∈ d, g(α) = P (j) where g is the witness to principle W added by

P, P is the surjection from ω1 to [ωV2 ]2 added by R0, and j = r(ot(d ∩ α)).

The ordering on R1 is end-extension. We note that since P ∗ Q ∗ R0 is proper, anordinal has cofinality ω in the extension by this poset if and only if it has cofinalityω in V .

Intuitively R1 is intended to add a club set in ωV2 of order type ω1 such that forevery pair (α, β) ∈ [ωV2 ]2 there are stationarily many points in that club set whichare mapped by g to the pair (α, β).

We need to know that there are sufficiently many conditions in R1. Accordinglywe prove that for every d ∈ R1, η < ω1 and ζ < ωV2 , if ot(r) ≤ η < ω1 then there ise ≤ d such that max(e) > ζ and ot(e) = η+ 1. We do this by induction on η for alld and ζ. Suppose we have established it for all η′ < η. We assume that η is limit,the successor case is similar and easier.

We will work in the extension by P ∗ Q ∗ R0. Let j = r(η), and P (j) = (α, β).Find a countable elementary substructure M of some large Hθ such that d, η, ζ ∈Mand γ = sup(M ∩ ωV2 ) is such that g(γ) = (α, β) (this is possible because we arein a proper extension of V and g assumes each value on a stationary set in ωV2 ).Now build (using the induction hypothesis) a decreasing sequence of conditions dnsuch that their union d∗ has order type η and supremum γ, and observe that byconstruction e = d∗ ∪ {γ} is a condition with order type η + 1.

Lemma 29. P ∗ Q ∗ R is proper.

Proof. We work in V . Let N ≺ Hθ be a countable elementary substructure con-taining everything relevant and let (p, q, r0, r1) be a condition in N . Let δ = N ∩ω1

and let γ = sup(N ∩ ω2).Let j = r(δ) so that j < δ and in particular j ∈ N . Extending (p, q, r0, r1)

to a stronger condition in N we may assume without loss of generality that (p, q)forces that j ∈ dom(r0) and determines that r0(j) = (α, β) for some α and β inN ∩ ω2 (so in particular α < β < γ). Since P is countably closed we may extend


p to a condition p∗ such that p∗ is strongly (N,P)-generic, the domain of p∗ is(γ + 1) ∩ cof(ω) and p∗(γ) = (α, β).

We then choose (q∗, r∗0) a name for an (N [GP],Q∗R0)-generic condition extending(q, r0), so that easily (p∗, q∗, r∗0) is an (N,P ∗ Q ∗ R0)-generic condition extending(p, q, r0). We force below this condition to get a generic filter G ∗H ∗ I, where weknow that N [G ∗H ∗ I] ∩ON = N ∩ON by the genericity of the condition.

We now build a decreasing sequence of length ω in N [G∗H ∗I]∩R1 which meetseach dense set inN [G∗H∗I]. Let r∗1 be the union of this sequence, where it is routineto check that sup(r∗1) = γ and ot(r∗1) = δ. We claim that r∗1 ∪ {γ} is a condition inR1. To see this we merely observe that r(δ) = j and P (j) = (α, β) = g(γ). ¤

Lemma 30. PFA holds in V P.

Proof. We apply the PFA to P ∗ Q ∗ R to get a suitably generic filter F . Taking theunion of the “P-parts” of the conditions in F we get a certain function p. Arrangingthat F meets suitable dense sets we can use the union of the “R-parts” to producea witness that p is a condition, and this is all we need. ¤

The upshot of this discussion is that we have proved the following theorem. Herethe consistency is relative to that of PFA. PFA is known to be consistent relativeto the existence of a supercompact cardinal.

Theorem 6. It is consistent that PFA holds and there is a 2-bounded colouring of[ω2]2 such that there is no closed polychromatic set of order type ω1.

8. A positive result from GCH

We saw in Section 2 that under CH there is a 2-bounded colouring of pairs fromω1 with no uncountable polychromatic set. A similar argument easily gives thatif κ<κ = κ for some uncountable cardinal κ, then there is a 2-bounded colouringof [κ]2 with no polychromatic set of size κ. We now prove a positive result about(< κ)-bounded colourings of [κ+]2 from the same assumption that κ<κ = κ.

Theorem 7. Let κ be an infinite cardinal and assume that κ<κ = κ. Then for every(< κ)-bounded colouring of [κ+]2 and every ordinal η < κ+ there is a polychromaticset of order type η.

Proof. Fix f a (< κ)-bounded colouring of [κ+]2. We may may assume without lossof generality that f is normal. Fix also η < κ+, where without loss of generalitywe may assume that η ≥ κ.

We will generalise an idea used earlier in the paper. If X ⊆ κ+ we define

F (X) =def {α : ∃γ ∃b ∈ [X]2 f(α, γ) = f(b)}Since f is normal, F (X) is the set of α such that there exist β, γ ∈ X with α, β < γand f(α, γ) = f(β, γ). Clearly F (X) ⊆ max(X). Since κ is regular we see that if|X| < κ, then |F (X)| < κ.

The proof of Lemma 3 goes through in this setting. so for all X,Y ⊆ κ+ andα < κ+, if X < α < Y , X ∪ Y and X ∪ {α} are polychromatic and α /∈ F (X ∪ Y ),then X ∪ {α} ∪ Y is polychromatic.

Now we will build a polychromatic set of order type η; actually for technicalreasons we will end up building the set with type η + 1. We start by fixing alarge regular θ and building a continuous increasing chain 〈Mi : i ≤ η〉 such that


Mi ≺ Hθ, |Mi| = κ, Mi ∩ κ+ ∈ κ+, Mi ∩ κ+ < Mi+1 ∩ κ+ for i < η, <κMi ⊆ Mi

for i successor.Let βi = Mi ∩ κ+ for i ≤ η, we note that i 7→ βi is strictly increasing and

continuous and also that cf(βi) = κ for i successor. Fix α > βη, and fix also abijection g : κ ' η. we will construct by induction ordinals αj for j < κ such that

(1) βg(j) ≤ αj < βg(j)+1.(2) {αk : k < j} ∪ {α} is polychromatic.

Suppose that we have chosen αk for k < j. Let X = {αk : k < j, g(k) < g(j)}and Y = {αk : k < j, g(k) > g(j)} ∪ {α}. Note that by construction X ⊆ βg(j) andY ⊆ κ+ \ βg(j)+1.

By the choice of the Mi we see that X ∈ Mg(j)+1. Since α > βg(j)+1 andX ∪ {α} is polychromatic, the usual elementarity argument shows that the setof α with X ∪ {α} polychromatic is unbounded in βg(j)+1. On the other hand|F (X ∪ Y )| < κ and cf(βg(j)+1) = κ, so that F (X ∪ Y ) is bounded in βg(j)+1.

So we may choose αj such that(1) βg(j) ≤ αj < βg(j)+1, in particular X < αj < Y .(2) X ∪ {αj} is polychromatic(3) αj /∈ F (X ∪ Y ).

Now we conclude that X ∪{αj}∪Y is polychromatic, so that the construction cancontinue. ¤

In particular under GCH the situation is now rather clear for κ+ with κ regularand uncountable: we have seen that if κ<κ = κ then κ+ →poly (α)2

<κ−bd for allα < κ+, while if 2κ = κ+ then κ+ 6→poly (κ+)2

2−bd.

Mathematics and Computer Science Department, Ben-Gurion University, Beer-Sheva,84105 Israel

E-mail address: [email protected]

Mathematical Sciences Department, Carnegie Mellon University, Pittsburgh PA15215, USA


Department of Mathematics, Massachusetts Institute of Technology, CambridgeMA 02139, USA


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