Infinite monochromatic paths and a theorem of Erdős-Hajnal-Rado

We prove that if μ is a singular cardinal with countable cofinality and 2μ = μ+ then ( μ+ μ ) 9 ( μ+ א2 μ μ ) . Mathematics Subject Classifications: 05C15, 05C63, 03E02

A central case is α = µ + , β = µ where µ is an infinite cardinal. We shall focus on the subcase in which µ is a singular cardinal and 2 µ = µ + . If κ is an infinite cardinal and 2 κ = κ + then κ + κ κ + κ 2 , and this holds at any infinite cardinal including singular cardinals as proved in [3]. We say that the strong polarized relation fails at the pair (κ, κ + ) under the local instance of GCH at κ.
These theorems show that we understand the balanced polarized relation quite well at successors of singular cardinals. A natural question concerns the intermediate unbalanced relation µ + µ → µ + α µ µ under the assumption 2 µ = µ + . This question has been investigated in [3], and the authors proved that if µ cf(µ) > ℵ 0 then 2 µ = µ + implies the negative relation µ + µ µ + ω µ µ . Namely, the unbalanced relation behaves much similarly to the strong polarized relation, yielding a negative statement under the assumption 2 µ = µ + already when α = ω in the second color. We indicate that the authors of [3] assume GCH for this result, but only the local instance at µ is needed for the proof.
Regarding the first part of the question we indicate that Jones proved in [7] that for every countable ordinal α, thus reaching well-nigh to ω 1 . Our main result here is a sharp negative answer to the second part. Namely, if µ > cf(µ) = ω then µ + µ µ + ω 2 µ µ under the assumption 2 µ = µ + , no matter how large is 2 ω . This includes, in particular, cases in which 2 µ = µ + but µ is not strong limit.
In order to prove our result we replace the Erdős-Rado theorem by a statement about monochromatic paths in complete graphs. We shall use a theorem of Todorčević from [15], see also [12]. It says that ω 2 → ipath (ω) 2 ω even if 2 ω ω 2 . On the other hand, ω 1 ipath (ω) 2 ω and even ω 1 ipath (ω) 2 ω,<ω as shown by Todorčević in [16]. Hence concerning the polarized relation under scrutiny, we do not resolve the focal case of ω 1 .
Along the refereeing process we learned that the above mentioned path relations were first proved by Todorčević. Our original manuscript contained similar statements with our proofs. Since the proofs are quite different (and some of the statements are not identical) we include a discussion on path relations in the current version. The paper is organized in such a way that the new result (concerning the polarized relation) appears in the first section, and path relations are discussed in the second, so the reader may skip that part. We are deeply indebted to the referees for pointing to the literature concerning path relations, and for many other helpful suggestions.
Our notation is mostly standard. We follow [4] with respect to arrows notation. Our set theoretical notation is coherent, in general, with [6], but we adopt the Jerusalem notation in forcing, so p P q reads p is weaker than q. An excellent background concerning the basics of the polarized relation can be found in [17].
( ) ω 1 ipath (ω) 2 ω,<ω . Our purpose in this section is to derive negative polarized relations at singular cardinals with countable cofinality from instances of the ordinary path relation. We shall need the following: Definition 3. Polarized relations with alternatives.
We say that It follows from the definition that if some relation holds with alternatives then it holds upon omitting one of the alternatives (or more). Of course, one may suggest an alternative only in one of the colors, as done in the following theorem which is the main result of this section.
We can address now the second part of Question 1 by eliminating the assumption 2 ω = ω 1 from the proof of µ + µ µ + ω 2 µ µ which appears in [3].
The problem of µ + µ → µ + ω 1 µ µ at singular cardinals with countable cofinality under the assumption 2 µ = µ + remains open. We believe that a positive relation is consistent. Moreover, in the light of [13] we even raise the possibility that it holds in ZFC under sufficiently strong assumptions of large cardinals: Question 6. Suppose that µ is an ω-limit of measurable cardinals. Is it provable that µ + µ → µ + ω 1 µ µ ?

Path relations the hard way
In this section we consider path relations at ω 2 and ω 1 . There is a slight difference between our concepts and the path relations of Scheepers and Todorčević mentioned in the previous section, as will be explicated anon. We emphasize that for the main result of the paper concerning polarized relations one can use the statements of Todorčević.
Path relations were considered in [14] and in [11], as well as in [2]. The latter has been continued in [1], [9] and [10]. Due to the terminology of [1] we distinguish two notions of paths. If the elements of the path are ordinals and they appear in increasing order then we shall say an increasing path. If the elements of the path are ordinals with no requirement of being increasing then we shall say a simple path. We use the notation κ → ipath (ω) 2 θ in the first case, and κ → path (ω) 2 θ in the second. Scheepers and Todorčević dealt with increasing paths, while the paths in this section are simple.
Definition 7. Simple path relations. Let θ and κ be cardinals.
Here is our first theorem: Theorem 8. Path relations at the second uncountable cardinal: ( ) If κ 2 ω then there is a coloring of the ordered pairs of κ with no monochromatic infinite path.
‫)ג(‬ For every coloring of the ordered pairs of ω 2 there is an infinite path which assumes at most two colors. Proof. Beginning with the first part, let c : [ω 2 ] 2 → ω be a coloring, and let χ be a sufficiently large regular cardinal. Choose an elementary submodel M ≺ (H(χ), ∈) of size ℵ 1 so that ω 1 ⊆ M and ω 1 , c ∈ M . Let δ = ω 2 ∩ M be the characteristic ordinal, so δ ∈ ω 2 and we choose M in such a way that cf(δ) = ω 1 . For every n ∈ ω let B n = {β ∈ δ : c(β, δ) = n}. Notice that δ = n∈ω B n , so there exists n 0 ∈ ω such that B n 0 is unbounded in δ. By induction on i ∈ ω we choose an ordinal β i ∈ δ such that the following requirements hold for every i ∈ ω: The choice is possible since B n 0 is unbounded in δ and since cf(δ) = ω 1 . For part (d) we use elementarity. Now the sequence (β 2i , β 2i+3 : i ∈ ω) forms a monochromatic path where consecutive ordinals are colored by n 0 , thus part (ℵ) has been proved.
For part ( ) fix an uncountable cardinal κ 2 ω . Let T be the full binary tree of height ω, and let (b α : α ∈ κ) enumerate κ-many distinct ω-branches of T . We shall define a coloring c over the ordered pairs of κ using ω + ω colors. Given two distinct ordinals α, β ∈ κ let m = m(α, β) ∈ ω be the departure level of b α and b β . Namely, Notice that if c(α, β) = m then c(β, α) = m 1− , so the order is crucial here.
We move to part ‫)ג(‬ which says that one can limit the above negative examples to two colors only along the path. To see this, choose M ≺ (H(χ), ∈) as in the proof of part (ℵ), and let δ, B n 0 be as there.
For every m ∈ ω let B m n 0 = {β ∈ B n 0 : c(δ, β) = m} and pick up m 0 ∈ ω for which B m 0 n 0 is unbounded in δ. By induction on i ∈ ω we choose β i as in the first part of the proof, but for each i ∈ ω we add the requirement that c(β 2i+1 , β 2j ) = m 0 for every j i. Elementarity guarantees that this is possible, recalling the definition of the set B m 0 n 0 . As before, the path will be (β i : i ∈ ω) and one can verify that {c(β i , β i+1 ), c(β i+1 , β i ) : i ∈ ω} = {m 0 , n 0 } as required.
If 2 ω = ω 1 then the above statements are immediate, so the main point of the above theorem is that it holds in ZFC. In particular, ω 2 → path (ω) 2 ω holds even if 2 ω ω 2 . The following theorem and corollary show that such a relation is impossible when ω 1 is deemed. Moreover, even weak homogeneity is excluded. We emphasize that this result follows from [16, 6.8], as pointed out by one of the referees. Our method of proof is to employ a forcing argument, and then to claim that it holds in ZFC by absoluteness.
In Lemma 10 below we shall prove that Q is ccc, so forcing with Q preserves cardinals. Let G ⊆ Q be V -generic. Define c = {f p : p ∈ G}. By the density of each D α we see that dom(c) = [ω 1 ] 2 . By the directness of G we see that c is a function. By the definition of the conditions we see that c exemplifies the required statement.
The coloring c has been forced, but we argue that such a coloring already exists in ZFC. To see this, notice that the existence of our coloring is expressible as an existence statement of a model of some formula ψ ∈ L ω 1 ω (Q) where Q is the quantifier of there exist uncountably many. Indeed, the statement of the theorem asserts that there is a coloring over an uncountable domain. This can be expressed using the quantifier Q. Now for every m ∈ ω we can express the property stated in the theorem by a first order formula. Using the infinitary logic L ω 1 ω we can form a conjuction of these statements for every m ∈ ω, so there is a formula ψ ∈ L ω 1 ω (Q) as required. By [8] we conclude that such a coloring exists in the ground model, so we are done.
Recall that a forcing notion Q is ccc iff |A| ℵ 0 whenever A is an antichain of Q. In order to accomplish the proof we need the following: Lemma 10. The forcing notion Q is ccc.
Proof. We commence with a general claim about projecting conditions in Q to a countable ordinal. Suppose that 0 < δ < ω 1 and δ is a limit ordinal. Suppose that q ∈ Q and let p = q δ, that is u p = u q ∩ δ and f p = f q [u p ] 2 . Notice that p = (u p , f p ) ∈ Q and p Q q.
Choose an increasing function h : u q → δ such that h u p is the identity function. Notice that h u q is bounded in δ since δ is a limit ordinal. We shall define a condition p [δ] as follows. First, let We argue that p [δ] q. Let us justify this statement and then explain how to derive the chain condition from it. Our purpose is to define a condition r so that q, p [δ] r. Let and f r [u q ] 2 = f q , upon noticing that on the common part of u p we assign the same values and on the rest we have disjoint sets so the information is not contradictory. It remains to define f r over mixed pairs, so assume that α ∈ u p [δ] − u p and β ∈ u q − u p . We let f r (α, β) = |u r | + 2 + |u r | · (|u r ∩ α| + |u r ∩ β|). Finally, let r = (u r , f r ).
It is clear that p [δ] , q r once we show that r ∈ Q, so we must prove that r satisfies the defining property of our conditions. Suppose, therefore, that v ⊆ {0, . . . , m}, {α : m} ⊆ u r and α : ∈ v is strictly increasing. We argue that |v| < max{f r (α , α +1 ) : < m}. In order to prove this, we consider three cases.
We are left with all the cases in which for every < m either {α , α +1 } ⊆ u p [δ] or {α , α +1 } ⊆ u q . Hence from now on we assume that Case 1 fails. Let us remark that it is fairly possible that for some k, < m we will have However, for this to happen we must have a non-monotonic sequence of αs, as we must fall back in u p in the middle of the process. Hence if v ⊆ {0, . . . , m} and α : ∈ v is increasing then necessarily {α : ∈ v} ⊆ u p [δ] or {α : ∈ v} ⊆ u q . We remain, therefore, with two cases: We wish to use the fact that u p [δ] is a condition and apply it to v. But the fact that {α : ∈ v} ⊆ u p [δ] does not guarantee that α ∈ u p [δ] for every m, so the assumption of the case is insufficient for this plain argument. Still, we can argue as follows. For each and we can assume that α / ∈ u q (otherwise α , α +1 ∈ u q , a case which was covered before), so α ∈ u p [δ] − u q . But then {α , α +1 } u p [δ] ∧ {α , α +1 } u q so this is covered in Case 1. If α ∈ u q and α +1 / ∈ u q we use a similar argument. So we conclude that < m ⇒ β = β +1 .
This is similar to the previous case.
We conclude, therefore, that r ∈ Q which shows that p [δ] and q are compatible. Having established this general claim let us prove the chain condition. Assume towards contradiction that {q α : α ∈ ω 1 } is an antichain in Q. For every limit ordinal α ∈ ω 1 let p α = q α ∩ α. Using Fodor's lemma we see that for some stationary subset S ⊆ ω 1 which consists of limit ordinals and a fixed condition p we have α ∈ S ⇒ p α = p. We shrink S further by assuming that all the conditions in {q α : α ∈ S} are pairwise isomorphic. It means that if |u qα | = n α for every α ∈ S then n α = n for some fixed n ∈ ω and for all the elements of S. Moreover, the order pattern of the elements of every u qα is assumed to be the same pattern for all the elements of S. Now choose ζ, η ∈ S so that ζ < η and u q ζ ⊆ η. Let δ be a limit ordinal such that p ⊆ δ. It follows that p ζ is the projection of p ζ to the countable ordinal δ. But then q ζ and q η are also compatible, a contradiction.
We can derive now our conclusion about path relations at ω 1 . We indicate that the above lemma fails while trying to apply the same forcing notion to higher cardinals. This might serve as an evidence to the possibility that the polarized relation µ + µ → µ + ω 1 µ µ for a singular cardinal with countable cofinality is consistent and maybe even provable under the assumption that 2 µ = µ + .