Strong colorings over partitions

A strong coloring on a cardinal $\kappa$ is a function $f:[\kappa]^2\to \kappa$ such that for every $A\subseteq \kappa$ of full size $\kappa$, every color $\gamma<\kappa$ is attained by $f\upharpoonright[A]^2$. The symbol $\kappa\nrightarrow [\kappa]^2_\kappa$ asserts the existence of a strong coloring on $\kappa$. We introduce the symbol $\kappa\nrightarrow_p[\kappa]^2_\kappa$ which asserts the existence of a coloring $f:[\kappa]^2\to \kappa$ which is strong over a partition $p:[\kappa]^2\to\theta$. A coloring $f$ is strong over $p$ if for every $A\in [\kappa]^\kappa$ there is $i<\theta$ so that every color $\gamma<\kappa$ is attained by $f\upharpoonright ([A]^2\cap p^{-1}(i))$. We prove that whenever $\kappa\nrightarrow[\kappa]^2_\kappa$ holds, also $\kappa\nrightarrow_p[\kappa]^2_\kappa$ holds for an arbitrary finite partition $p$. Similarly, arbitrary finite $p$-s can be added to stronger symbols which hold in any model of ZFC. If $\kappa^\theta=\kappa$, then $\kappa\nrightarrow_p[\kappa]^2_\kappa$ and stronger symbols, like $\mathrm{Pr}_1(\kappa,\kappa,\kappa,\chi)$ or $\mathrm{Pr}_0(\kappa,\kappa,\kappa,\aleph_0)$, hold also for an arbitrary partition $p$ to $\theta$ parts.


Introduction
The theory of strong colorings branched off Ramsey Theory in 1933 when Sierpinski constructed a coloring on [R] 2 that contradicted the uncountable generalization of Ramsey's theorem. For many years, pair-colorings which keep their range even after they are restricted to all unordered pairs from an arbitrary, sufficiently large set were called "bad"; now they are called "strong". Definition 1. Let λ ≤ κ be cardinals. A strong λ-coloring on κ is a function f : By Ramsey's theorem there are no strong λ-dolorings on ω for λ > 1. Sierpinski constructed a strong 2-coloring on the continuum and on ℵ 1 .
Assertions of existence of strong colorings with various cardinal parameters are conveniently phrased with partition-calculus symbols. The (negative) square-brackets symbol κ [κ] 2 λ , asserts the existence of a strong λ-coloring on κ. Recall that the symbol for Ramsey's theorem for pairs, 2000 Mathematics Subject Classification. Primary: 03E02, 03E17, 03E35. Secondary: 03E50. Key words and phrases. Strong coloring, Ramsey theory, Generalized Continuum Hypothesis, Forcing, Martin Axiom.
The first autor's research for this paper was partially supported by an Israeli Science Foundation grant number 665/20.
The third author's research for this paper was partially supported by NSERC of Canada.
reads "for every f : [ω] 2 → n there is an infinite subset A ⊆ ω such that f ↾ [A] 2 is constant (omits all colors but one)". The square brackets in place of the rounded ones stand for "omits at least one color"; with the negation on the arrow, the symbol κ [κ] 2 λ means, then, "not for all colorings f : [κ] 2 → λ at least one color can be omitted on [A] 2 for some A ⊆ κ of cardinality |A| = κ". That is, there exists a strong λ-coloring on κ. When 2 is replaced with some d > 0 the symbol states the existence of an analogous coloring of unordered d-tuples. As Ramsey's theorem holds for all finite d > 0, strong d-dimensional colorings can also exist only on uncountable cardinals. In what follows we shall address almost exlusively the case d = 2.
Definition 2. Given a coloring f : The collection of f -strong subsets of [κ] 2 is clearly upwards closed and not necessarily closed under intersections.
Let us conclude the introduction with the remark that some authors use the term "strong coloring" only for colorings which witness Pr 1 or a stronger symbol.

A brief history of strong colorings
Strong κ-colorings on various cardinals κ were constructed by Erdős, Hajnál, Milner and Rado in the 1950's and 1960's from instances of the GCH. For every cardinal κ they were able to construct from 2 κ = κ + colorings f : and even colorings which witnesses the stronger [5]). A coloring f [κ + ] 2 → κ + witnesses this symbol if and only if for every B ∈ [κ + ] κ + , for all but fewer than κ ordinals α < κ + the full range κ + is attained by f on the set Galvin [15], who was motivated by the problem of productivity of chain conditions and by earlier work of Laver, used 2 κ = κ + to obtain a new class of 2-colorings, which in modern notation witness Pr 1 (κ + , κ + , 2, ℵ 0 ), and used these colorings for constructing counter examples to the productivity of the κ + -chain condition. A straightforward modification of Galvin's proof actually gives Pr 1 (κ + , κ + , κ + , ℵ 0 ) on all successor cardinals from 2 κ = κ + .
Ramsey's theorem prohibits the existence of strong colorings (with more than one color) on countable sets for which all infinite subsets are strong, but in topological partition theory, strong colorings may exist also on countable spaces. Baumgartner [3], following some unpublished work by Galvin, constructed a coloring c : [Q] 2 → ω which attains all colors on every homeomorphic copy of Q. Todorčević [45] obtained the rectangular version of Baumgartner's result and very recently, Raghavan and Todorčević [22] proved that if a Woodin cardinal exists then for every natural number k > 2, for every coloring c : [R] 2 → k there is homeomorphic copy of Q in R on which at most 2 colors occur, confirming thus a conjecture of Galvin from the 1970s. They also proved that any regular topological space of cardinality ℵ n admits a coloring of (n + 2)-tuples which attains all ω colors on every subspace which is homeomorphic to Q.

Strong-coloring symbols over partitions
We introduce now the main new notion of symbols with an additional parameter p, where p is a partition of unordered pairs. Suppose p : [κ] 2 → θ is a partition of unordered pairs from κ. A preliminary definition of the square brackets symbol κ p [κ] 2 κ with parameter p has been mentioned in the abstract: there exists a coloring f : However, for Pr 1 or for Pr 0 it is not possible to require a prescribed pattern on a ⊛ b in both f and p when a, b belong to an arbitrary A, as all such a ⊛ b might meet more than one p-cell. What we do, then, is replace this definition by a different one. The new definition is equivalent to the initial definition in all square-bracket symbols by Fact 5 below, and works for Pr 1 and Pr 0 . (1) For a function ζ : θ → λ and α ∈ [κ] d we say that f hits ζ over p at α, if f (α) = ζ(p(α)).
Thus, the initial definition of an (f, p)-strong X ⊆ [κ] d -that (X ∩ p −1 (i)) is f -strong for some fixed p-cell i -is replaced in (2) above with the requirement that every assigment of colors to p-cells ζ : θ → λ is hit by some d ∈ X. The advantage of the new definition is that an assignment ζ can be hit in any p-cell, so defining Pr 1 and Pr 0 over a partition will now make sense.
The definitions of the main symbols over partitions which we shall work with are in Definition 7 below; an impatient reader can proceed there directly. We precede this definition with two useful facts about (f, p)-strong sets.
For the other direction suppose to the contrary that for every i < λ there is some A simple book-keeping argument can waive the dependence of h p on p for a set of ≤ λ <µ partitions: Proof. Suppose h : [κ] d → λ <µ and p = p δ : δ < λ <µ are given, where p δ : [κ] d → θ δ and θ δ < µ for every δ < λ <µ .
We define now the main symbols over a partition. We state only the case for pairs. The definitions of the square-bracket symbols for d = 2 are similar. (1) The symbol asserts the existence of a coloring f : (3) The symbol Pr 1 (κ, µ, λ, χ) p asserts the existence of a coloring f : [κ] 2 → λ such that for every ξ < χ and a family A ⊆ [κ] <ξ of pairwise disjoint nonempty subsets of κ such that |A| = µ, for Pr 0 (κ, µ, λ, χ) p asserts the existence of a coloring f : are the i th and j th elements of a and of b, respectively, in increasing order.
In each of the four symbols above, writing p instead of p means there exists a single coloring which witnesses simultaneously the relation with p δ in place of p for each δ < δ( * ).
By Fact 5, the first two symbols are equivalently defined by requiring that for every Then every coloring f which witnesses λ . In particular, Given some ζ ∈ λ θ , fix a = {α, β} and b = {γ, δ} from A such that α < β < γ < δ and such that f hits ζ over p at all (four) elements {x, y} ∈ a ⊛ b. In particular, f hits ζ over p at {α, δ} which belongs to A ⊛ B.
The next lemma is the main tool for adding a partition parameter to a strong-coloring symbol.

Valid symbols over partitions in ZFC and in ZFC with additional axioms
Question 10. Suppose κ ≥ ρ are cardinals. Which strong-coloring symbols in κ hold over all < ρ partitions?
Clearly, every coloring which witnesses a strong-coloring symbol Φ over some partition p, witnesses the symbol gotten by deleting p from Φ. The question of existence of strong colorings over partition therefore refines the question of existence of strong cvolorings in the classical sense.
Let us state ZFC symbols over partitions whose classical counterparts were mentioned in Section 2 above: Theorem 12. For every regular cardinal κ and a sequence of length κ + of finite partitions The symbol without p holds by the results of Todorčević, Moore and Shelah. Now apply Lemma 9(1).
In particular, ω holds by Todorevic's [44], and now apply Lemma 6 as in the proof of Lemma 9.
Lastly in this section, we show that | • (κ), an axiom (stated in the proof below), which does not imply 2 κ = κ + , implies the following rectangular square-brackets symbol.
In this Section we shall show that the existence of strong colorings over countable partitions of [ω 1 ] 2 is independent over ZFC and over ZFC + 2 ℵ 0 > ℵ 1 .
Before proving yet another combinatorial property in a Cohen extension let us recall Roitman's [31] proof that the addition of a single Cohen real introduces an S-space, Todorčević's presentation in [43], p. 26 and Rinot's blog-post [27] in which it is shown that a single Cohen real introduces Pr 0 (ℵ 1 , ℵ 1 , ℵ 0 , ℵ 0 ). For a short proof of Shelah's theorem that a single Cohen real introduces a Suslin line see [41]. Fleissner [14] proved that adding λ Cohen reals introduces two ccc spaces whose product is not λ-cc. Hajnal and Komjath [17] proved that adding one Cohen subset to a cardinal κ = κ <κ forces the statement Q(κ + ) they defined, following [4]: for every graph G = κ + , E with χ(G) = κ + there is a coloring f : E → κ + such that for every partition of κ + to κ parts, all colors are gotten by f on edges from a single part. It is still open if Q(ℵ 1 ) holds in ZFC.
Theorem 27. If C ℵ 2 is the partial order for adding ℵ 2 Cohen reals then for every sequence p = p δ : δ < ω 1 of partitions p δ : [ω 1 ] 2 → ω in the forcing extension by C ℵ 2 , Proof. Let C α be the partial order of finite partial functions from [α] 2 to ω. Let V be a model of set theory and let G ⊆ C ω 2 be generic over V . Then G : [ω 2 ] 2 → ω. Now suppose that p = p δ : δ < ω 1 is an arbitrary sequence of partitions p δ : In V , fix a sequence e α : ω ≤ α < ω 1 , where e α : ω → α is a bijection. In the generic extension, define a coloring f : for β ≥ ω and as 0 otherwise.
The forcing for adding a single Cohen real is obviously σ-linked. Thus, the next theorem applies to a broader class of posets than Cohen forcing. The previous theorem holds also in this generality.
Corollary 29. It is consistent with MA ℵ 1 (σ-linked) that Pr 0 (ℵ 1 , ℵ 1 , ℵ 1 , ℵ 0 )p holds for any ω 1 sequence of partitionsp = {p ξ } ξ∈ω 1 such that p ξ : [ω 1 ] 2 → ω. Now we prove that the symbol can consistently fail for some p : We actually prove more. The failure of the symbol above over a partition p : [ω 1 ] 2 → ω, symbolically written as ) omits at least one color for every i < ω. Let us introduce the following symbol: ω 1 → p [ω 1 ] 2 ω 1 \ω 1 , to say that for every coloring f : [ω 1 ] 2 → ω 1 there is a set A ∈ [ω 1 ] ℵ 1 such that for every i < ω a set of size ℵ 1 of colors is omitted by f ↾ ([A] 2 ∩ p −1 (i)). An even stronger failure (via breaking ω 1 to two disjoint equinumerous sets and identifying all colors in each part) is It is the consistency of the latter symbol which we prove. Note that with the roundedbrackets symbol in (1) from the introduction we may write this failure as: whose meaning is that for every coloring f : Thus, while ω 1 [ω 1 ] 2 ω 1 holds in ZFC, it is consistent that for a suitable countable partition p the symbol ω 1 p [ω 1 ] 2 ω 1 fails pretty badly.
Theorem 30. It is consistent that 2 ℵ 0 = ℵ 2 and there is a partition p : Proof of the theorem. Let P be the partial order of finite partial functions from [ω 1 ] 2 → ω ordered by inclusion. More precisely, each condition q ∈ P has associated to it a finite subset of ω 1 which, abusing notation, will be called dom (q). Then q is a function [dom (q)] 2 → ω.
It suffices to establish the following two claims.
Claim 33. The partial order Q ω 2 satisfies the ccc.
This completes the proof of the Theorem.
Note that for µ ≤ λ this symbol is stronger than κ → p [κ] 2 λ\λ . Thus the next theorem, which uses ideas from [40], gives a stronger consistency than the previous one.
Theorem 35. Given any regular κ > ℵ 1 it is consistent that: ω,<ω . Theorem 36. Given any regular κ > ℵ 1 it is consistent that: The proofs of both theorems are similar, using ideas from [40]; only the proof of Theorem 35 will be given in detail. Both rely on the following definition: Definition 37. Let µ be some probability measure on ω under which each singleton has positive measure, for example µ({n}) = 2 −n . A sequence of functions P = {p η } η∈ω 1 will be said to have full outer measure if: • p η : η → ω • for each η ∈ ω 1 the set {p β ↾ η} β>η has measure one in the measure space (ω η , µ η ).
By enumerating all functions from a countable ordinal into ω, we have: Assuming the Continuum Hypothesis there is a sequence P = {p η } η∈ω 1 such that {p β ↾ η} β>η = ω η for each η ∈ ω 1 . Hence P has full outer measure as in Definition 37.
While it is, of course, impossible to preserve the property that {p β ↾ η} β>η = ω η when adding reals, the goal of the following arguments is to show that the properties of Definition 37 can be preserved in certain circumstances. The following definition is from [40] and will play a key role in this context. Definition 39. A function ψ : ω <ω → [ω 1 ] <ℵ 0 satisfying that ψ(s) ∩ψ(t) = ∅ unless s = t will be said to have disjoint range. If for each t ∈ ω <ω there is k such that |ψ(t ⌢ j)| < k for all j ∈ ω then ψ will be called bounded with disjoint range. If G is a filter of subtrees of ω <ω and ψ has disjoint range define If G is a generic filter of trees over a model V define It is shown in [40] that Lemma 40 and Lemma 42 hold.
Lemma 41 is the content of §3 of [1]. Recall that if I is an ideal then X is said to be orthogonal to I if X ∩ A is finite for each A ∈ I.
Lemma 41 (Abraham and Todorčević). Let I be a P-ideal on ω 1 that is generated by a family of ℵ 1 countable sets and such that ω 1 is not the union of countably many sets orthogonal to I. Then there is a proper partial order P I , that adds no reals, even when iterated with countable support, such that there is a P I -nameŻ for an uncountable subset of ω 1 such that 1 P I "(∀η ∈ ω 1 )Ż ∩ η ∈ I".
Lemma 43. Let P be a sequence with full outer measure and suppose that p = p(P). Suppose further that Then there is an uncountable Proof. In V [G] let L = G be the Laver real. In V [G][H] let R be the uncountable set given by Lemma 42. Construct by induction distinct ρ ξ ∈ R such that if η ∈ ξ then L(p(ρ ξ , ρ η )) > c(ρ ξ , ρ η ). To carry out the induction assume that R η = {ρ ξ } ξ∈η have been chosen and satisfy the inductive hypothesis. By the choice of R it follows that R η ∈ S b (G). Since P S b (Ġ) adds no new reals it follows that R η ∈ V [G] and so there is T ∈ G and ψ ∈ V with bounded, disjoint range such that T L "Ṙ η = S(Ġ, ψ)". Let µ be so large that T L "Ṙ η ⊆ µ" and let r be the root of T . For t ∈ T define W t = {x ∈ 2 µ | x ↾ ψ(t) has constant value |t|} and then define x ∈ W s } . Note that W + t has measure one in 2 µ for each t ⊇ r. To see this note that for a random h ∈ 2 µ the probability that h(ζ) = |t| + 1 is 2 −(|t|+1) . Also, note that since ψ is bounded -see Definition 39 -there is some k such that |ψ(s)| ≤ k for each s ∈ succ T (t). Hence, the probability of h belonging to W s is bounded below by 2 −(|t|+1)k for all s ∈ succ T (t) and these events are independent because the ψ(s) are pairwise disjoint for s ∈ succ T (t).
Define f on j≤|r| ψ(r ↾ j) to have constant value |r| and note that the domain of f is disjoint from each ψ(s) where s r. Hence the probability that f ⊆ h is non-zero and independent from belonging to each W + . Since p has full outer measure it follows that is uncountable and belongs to V [G]. Therefore by Lemma 42 there is some β ∈ R \ R η such that f ⊆ p β ↾ µ and such that for all t ∈ T containing r there are infinitely many s ∈ succ T (t) such that p(α, β) = |s| for all α ∈ ψ(s).
Using this and the definition of f , it is possible to start with r and successively thin out the successors of each t ∈ T to find a tree T * ⊆ T with root r such that p(α, β) = |t| for all t ∈ T * and for all α ∈ ψ(t). Once again starting with r and removing only finitely many elements of succ T * (t) for each t ∈ T * it is possible to find T * * ⊆ T * with root r such that (∀t ∈ T * * )(∀s ∈ succ T * * (t))(∀α ∈ ψ(t)) s(|t|) = s(p(α, β)) > c(α, β) and this implies that T * * L "(∀α ∈Ṙ η )L(p(α, β)) > c(α, β)". Since this holds for any T , genericity yields that in V [G][H] there is some β ∈ R \ R η such that L(p(ρ ξ , β)) > c(ρ ξ , β) for each ξ ∈ η. Define ρ η = β to continue the induction. Since limit stages are immediate, this completes the proof.
Proof of Theorem 35. The required model is the one obtained by starting with a model of the Continuum Hypothesis in which 2 ℵ 1 = κ. Then iterate with countable support the partial order L * P S bĠ ) . In the initial model there is, by Proposition 38, a sequence with full outer measure. To see this, begin by observing that it is shown in Theorem 7.3.39 of [2] that L preserves ⊑ Random . Since P S(Ġ) is proper and adds no new reals it is immediate that it also preserves ⊑ Random . It follows by Theorem 6.1.13 of [2] that the entire countable support iteration preserves outer measure sets and, hence, any sequence with full outer measure in the initial model maintains this property throughout the iteration.
To see that for every function c : [ω 1 ] 2 → ω there is an uncountable set witnessing ℵ 1 → p [ℵ 1 ] ℵ 0 ,<ℵ 0 use Lemma 3.4 and Lemma 3.6 of [1] to conclude that each partial order in the ω 2 length iteration is proper and has the ℵ 2 -pic of Definition 2.1 on page 409 of [36]. By Lemma 2.4 on page 410 of [36] it follows that the iteration has the ℵ 2 chain condition and, hence, that c appears at some stage. It is then routine to apply Lemma 43.
That b = ℵ 2 is a standard argument using that Laver forcing adds a dominating real.
Remark 44. The proof of Theorem 36 is similar but uses Miller reals instead of Laver reals. This requires that nowhere meagreness play the role of full outer measure.
Remark 45. Note that there is no partition p such that because a colouring c : [ω 1 ] 2 → ω 1 that is a bijection will provide a counterexample.

Concluding Remarks and Open Questions
It turns out, via Lemma 9, that getting strong coloring symbols over finite partitions is not harder than getting them without partititions; so one immediately gets many strong coloring symbols over partitions outright in ZFC. If the number of colors λ raised to the number of cells in a partition is not too large, Lemma 9 applies again, and consequently all GCH symbols gotten by Erdős, Hajnal and Milner on κ + hold under the GCH over arbirary κ-partitions. Even without instances of the GCH, strong colorings symbols over countable partitions are valid in Cohen-type forcing extenstions, by Theorems 27 and 28.
Yet, it is not the case that every time a strong-coloring symbol holds at a successor of a regular, it also holds over countable partitions: by Theorem 30 and 35 the ZFC symbol ℵ 1 [ℵ 1 ] 2 ℵ 1 , and hence all stronger ones, consistently fail quite badly over sufficiently generic countable partitions. Thus, strong coloring symbols over partitions are a subject of their own, in which the independence phenomenon is manifested prominently.
Many natural questions about the combinatorial and set-theoretic connections between coloring and partition arise. We hope that this subject will get attention in the near future both in the infinite combinatorics and in the forcing communities. For example, by Fact 11, there is always a set of 2-partitions of [κ + ] 2 such that no coloring is strong over all of them. What is the least cardinality of such a set? In the case of θ = κ = ℵ 0 , the results in Section 5 show that this cardinal may be as small as 1 or at least as large as ℵ 2 = κ ++ . Can this number ever be κ or, say, κ + < 2 κ ?
We conclude with a short selection of open questions.
Without partitions, both implications above hold.
Question 49. Is it consistent that there is a partition p such that for some integer k?
The consistency of this symbol is open even without the p. A negative answer may be easier to get with p.