PARTITIONING A REFLECTING STATIONARY SET
MAXWELL LEVINE AND ASSAF RINOT
Abstract. We address the question of whether a reflecting stationary set may be parti- tioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinals combinatorics, we infer that it is never the case that there exists a singular cardinal all of whose scales are very good.
1. Introduction A fundamental fact of set theory is Solovay’s partition theorem [Sol71] asserting that, for every stationary subset 푆 of a regular uncountable cardinal 휅, there exists a partition ⟨푆푖 | 푖 < 휅⟩ of 푆 into stationary sets. The standard proof involves the analysis of a certain 퐶-sequence over a stationary subset of 푆; such a sequence exists in ZFC (cf. [LS09]), but assuming the existence of better 퐶-sequences, stronger partition theorems follow. For instance: + ∙ (folklore) If 휅 = 휆 and 휆 holds, then any stationary subset 푆 of 휅 may be par- titioned into non-reflecting stationary sets ⟨푆푖 | 푖 < 휅⟩. That is, for all 푖 < 휅, 푆푖 is stationary, but Tr(푆푖) := {훽 < sup(푆푖) | cf(훽) > 휔 & 푆푖 ∩ 훽 is stationary in 훽} is empty. ∙ [Rin14, Lemma 3.2] If (휅) holds, then for every stationary 푆 ⊆ 휅, there exists a ⃗ coherent 퐶-sequence 퐶 = ⟨퐶훼 | 훼 < 휅⟩ such that 푆푖 := {훼 ∈ 푆 | min(퐶훼) = 푖} is ⃗ stationary for all 푖 < 휅. The coherence of 퐶 implies that the elements of ⟨Tr(푆푖) | 푖 < 휅⟩ are pairwise disjoint as well. ∙ [BR19, Theorem 1.24] If (휅) holds, then any fat subset of 휅 may be partitioned into 휅-many fat sets that do not simultaneously reflect. This raises the question of whether there is a fundamentally different way to partition large sets. A more concrete question reads as follows: Question 1.1. Suppose that 푆 is a subset of a regular uncountable cardinal 휅 for which ⋂︀ Tr(푆) is stationary. Can 푆 be split into sets ⟨푆푖 | 푖 < 휃⟩ in such a way that 푖<휃 Tr(푆푖) is stationary? And how large can 휃 be?
Remark. Note that for any sequence ⟨푆푖 | 푖 < 휃⟩ of pairwise disjoint subsets of 휅, the ⋂︀ 휅 intersection 푖<휃 Tr(푆푖) is a subset of 퐸≥휃. Therefore, the only cardinals 휃 of interest are the ones for which 휅 ∖ 휃 still contains a regular cardinal. The above question leads us to the following principle:
Date: February 15, 2019. The second author was partially supported by the European Research Council (grant agreement ERC- 2018-StG 802756) and by the Israel Science Foundation (grant agreement 2066/18). 1 2 MAXWELL LEVINE AND ASSAF RINOT
Definition 1.2. Π(푆, 휃, 푇 ) asserts the existence of a partition ⟨푆푖 | 푖 < 휃⟩ of 푆 such that ⋂︀ 푇 ∩ 푖<휃 Tr(푆푖) is stationary. In Magidor’s model [Mag82, S2], Π(푆, ℵ , 푇 ) holds for any two stationary subsets 푆 ⊆ 퐸ℵ2 1 ℵ0 and 푇 ⊆ 퐸ℵ2 , and it is also easy to provide consistent affirmative answers to Question 1.1 ℵ1 without appealing to large cardinals. However, the focus of this short paper is to establish that instances of the principle Π(...) hold true in ZFC. It is proved: Theorem A. Suppose that 휇 < 휃 < 휆 are infinite cardinals, with 휇, 휃 regular. (1) If 휆 is inaccessible, then Π(휆, 휃, 휆) and Π(휆+, 휆, 휆+) both hold; 휆+ 휆+ (2) If 휆 is regular, then Π(퐸휇 , 휃, 퐸휃 ) holds; 휃 휆+ 휆+ (3) If 2 ≤ 휆 and 휃 ̸= cf(휆), then Π(퐸휇 , 휃, 퐸휃 ) holds; 휆+ 휆+ (4) If 휆 is singular, then Π(퐸휇 , 휃, 퐸≤휃+3 ) holds. It is worth mentioning that our proof of Clause (4) is indeed fundamentally different than all standard proofs for partitioning a stationary set. We build on the fact that any singular cardinal admits a scale and that the set of good points of a scale is stationary relative to any cofinality; we also use a combination of club-guessing with Ulam matrices to avoid any cardinal arithmetic hypotheses. We initiated this project since we realized that ZFC instances of Π(...) would allow us to prove that the statement “all scales are very good” is inconsistent. And, indeed, the following is an easy consequence of Theorem A: Corollary. Suppose that 휆 is a singular cardinal, and ⃗휆 is a sequence of a regular cardinals of length cf(휆), converging to 휆. If ∏︀ ⃗휆 carries a scale, then it also carries a scale which is not very good. In this short paper, we also consider a simultaneous version of the principle Π(...) which is motivated by a simultaneous version of Solovay’s partition theorem recently obtained by Brodsky and Rinot:
Lemma 1.3 ([BR19, Lemma 1.15]). Suppose that ⟨푆푖 | 푖 < 휃⟩ is a sequence of stationary ′ subsets of a regular uncountable cardinal 휅, with 휃 ≤ 휅. Then there exists a sequence ⟨푆푖 | 푖 ∈ 퐼⟩ of pairwise disjoint stationary sets such that: ′ ∙ 푆푖 ⊆ 푆푖 for every 푖 ∈ 퐼; ∙ 퐼 is a cofinal subset of 휃.1 Evidently, Solovay’s theorem follows by invoking the preceding theorem with a constant sequence of length 휃 = 휅. The simultaneous version of Definition 1.2 reads as follows. ∐︀ Definition 1.4. (푆, 휈, 푇 ) asserts that for every 휃 ≤ 휈, every sequence ⟨푆푖 | 푖 < 휃⟩ of ′ ⋂︀ ′ subsets of 푆 and every stationary 푇 ⊆ 푇 ∩ 푖<휃 Tr(푆푖), there exists a sequence ⟨푆푖 | 푖 ∈ 퐼⟩ of pairwise disjoint stationary sets such that: ′ ∙ 푆푖 ⊆ 푆푖 for every 푖 ∈ 퐼; 1Note that if we demand that 퐼 be equal to 휃, then we do not get a theorem in ZFC. PARTITIONING A REFLECTING STATIONARY SET 3
∙ 퐼 is a cofinal subset of 휃; ′ ⋂︀ ′ ∙ 푇 ∩ 푖∈퐼 Tr(푆푖) is stationary. It is proved: Theorem B. Suppose that 휈 < 휆 are uncountable cardinals with 휈 ̸= cf(휆), and 2휈 ≤ 휆. ∐︀ + 휆+ Then, any of the following implies that (휆 , 휈, 퐸휈 ) holds: (1) 휆 is a regular cardinal; (2) 휆 is a singular cardinal admitting a very good scale.
휅 휅 Notation and conventions. For cardinals 휃 < 휅, we let 퐸휃 := {훼 < 휅 | cf(훼) = 휃}; 퐸̸=휃, 휅 휅 휅 + 퐸>휃, 퐸≥휃 and 퐸≤휃 are defined similarly. For a set of ordinals 푎, we write acc (푎) := {훼 < sup(푎) | sup(푎 ∩ 훼) = 훼 > 0} and acc(푎) := 푎 ∩ acc+(푎). The class of all ordinals is denoted by ORD. We also let Reg(휅) := {휃 < 휅 | ℵ0 ≤ cf(휃) = 휃}.
2. pcf scales In this section, we recall the notion of a scale for a singular cardinal (also known as a pcf scale) and the classification of points of a scale. These concepts will play a role in the proof of Theorems A and B. In turn, we also present an application of the partition principle Π(...) to the study of very good scales. ⃗ Definition 2.1. Suppose that 휆 is a singular cardinal, and 휆 = ⟨휆푖 | 푖 < cf(휆)⟩ is a strictly increasing sequence of regular cardinals, converging to 휆. For any two functions 푓, 푔 ∈ ∏︀ ⃗휆 and 푖 < cf(휆), we write 푓 <푖 푔 iff, for all 푗 ∈ cf(휆) ∖ 푖, 푓(푗) < 푔(푗). We also write 푓 <* 푔 to expresses that 푓 <푖 푔 for some 푖 < cf(휆).
⃗ + Definition 2.2. Suppose that 휆 is a singular cardinal. A sequence 푓 = ⟨푓훾 | 훾 < 휆 ⟩ is said to be a scale for 휆 iff there exists a sequence ⃗휆 as in the previous definition, such that: + ∏︀ ⃗ ∙ for every 훾 < 휆 , 푓훾 ∈ 휆; + * ∙ for every 훾 < 훿 < 휆 , 푓훾 < 푓훿; ∏︀ ⃗ + * ∙ for every 푔 ∈ 휆, there exists 훾 < 휆 such that 푔 < 푓훾. An ordinal 훼 < 휆+ is said to be: ∙ good with respect to 푓⃗ iff there exist a cofinal subset 퐴 ⊆ 훼 and 푖 < cf(휆) such that, 푖 for every pair of ordinals 훾 < 훿 from 퐴, 푓훾 < 푓훿; ∙ very good with respect to 푓⃗ iff there exist a club 퐶 ⊆ 훼 and 푖 < cf(휆) such that, for 푖 every pair of ordinals 훾 < 훿 from 퐶, 푓훾 < 푓훿. We also let: ⃗ 휆+ ⃗ ∙ 퐺(푓) := {훼 ∈ 퐸̸=cf(휆) | 훼 is good with respect to 푓}; ⃗ 휆+ ⃗ ∙ 푉 (푓) := {훼 ∈ 퐸̸=cf(휆) | 훼 is very good with respect to 푓}.
휆+ ⃗ ⃗ ⃗ Clearly, 퐸 Evidently, 푓훾 and 푔훾 differ on at most a single index, and so ⃗푔 is a scale. However, ⃗푔 fails to be very good at any given point 훼 ∈ 푇 . To see this, fix an arbitrary club 퐶 ⊆ 훼 and an index 푖 < cf(휆). Let 훾 := min(퐶 ∩ 푆푖) and 훿 := min(퐶 ∩ 푆푖 ∖ (훾 + 1)). Then 훾 < 훿 is a pair of elements of 퐶, while 푔훾(푖) = 0 = 푔훿(푖). Remark 2.6. Gitik and Sharon [GS08] constructed a model in which ℵ휔2 carries a very good scale in one product and a bad (i.e., not good) scale in another — hence , let alone ℵ휔2 푉 = 퐿, cannot hold. The question, then, is whether the former product carries only very good scales. Our results show that it does not, and in fact that it is a theorem of ZFC that in any product that carries a scale, there are scales which are not very good. Furthermore, it follows from the preceding proof together with Corollary 3.10 below (using 휃 := cf(휆)) that 2For an excellent survey, see [Eis10]. PARTITIONING A REFLECTING STATIONARY SET 5 any scale for a singular cardinal 휆 may be manipulated to have its set of very good points to not be a club relative to cofinality 휈 for unboundedly many regular cardinals 휈 < 휆. 3. Theorem A Recall that 풟(휈, 휃) stands for cf([휈]휃, ⊇),3 i.e., the least size of a family 풟 ⊆ [휈]휃 with the property that for every 푎 ∈ [휈]휃, there is 푑 ∈ 풟 with 푑 ⊆ 푎. Suppose now that 휈 is regular and uncountable; we let 풞(휈, 휃) denote the least size of a family 풞 ⊆ [휈]휃 with the property that for every club 푏 in 휈, there is 푐 ∈ 풞 with 푐 ⊆ 푏. It is well-known that 풞(휈, 휈), better known as cf(NS휈, ⊆), can be arbitrarily large. In contrast, for small values of 휃, 풞(휈, 휃) is provably small: Lemma 3.1. Suppose that 휃 ≤ 휈 are cardinals, with 휈 regular uncountable. Then: ∙ 휈 ≤ 풞(휈, 휃) ≤ 풟(휈, cf(휃)) ≤ 2휈; ∙풞 (휈, 휃) = 휈 whenever 휃++ < 휈; ∙풞 (휈, 휃) = 휈 whenever 휃 < cf(휃)+ < 휈. Proof. Evidently, 풟(휈, cf(휃)) ≤ 휈cf(휃) ≤ 휈휈 = 2휈. As, for all 훼 < 휈, 휈 ∖ 훼 is a club in 휈, we also have 휈 ≤ 풞(휈, 휃). In addition, it is obvious that if cf(휃) = 휃 then 풞(휈, 휃) ≤ 풟(휈, cf(휃)). Next, suppose that 휇 is an arbitrary infinite regular cardinal, with 휇+ < 휈. By Shelah’s 휈 club-guessing theorem [She94, III.S2], there exists a sequence ⟨푐훼 | 훼 ∈ 퐸휇⟩ having the 휈 following crucial property: for every club 푐 in 휈, there exists 훼 ∈ 퐸휇 such that 푐훼 ⊆ 푐 ∩ 훼 휈 and otp(푐훼) = 휇. It follows that {푐훼 | 훼 ∈ 퐸휇} witnesses that 풞(휈, 휇) ≤ 휈. In particular: ∙ if 휃++ < 휈, then using 휇 := 휃+. we have 풞(휈, 휃) ≤ 풞(휈, 휇) = 휈; ∙ if 휃 < cf(휃)+ < 휈, then using 휇 := 휃, we have 풞(휈, 휃) = 풞(휈, 휇) = 휈. Finally, we are left with dealing with the case that 휈 ∈ {휃+, 휃++} and cf(휃) < 휃. For every 휇 ∈ Reg(휃), fix a family 풞휇 witnessing that 풞(휈, 휇) = 휈. Fix an enumeration {푐훿 | ⋃︀ 훿 < 휈} of 휇∈Reg(휃) 풞휇. Also, fix a family 풟 witnessing the value of 풟(휈, cf(휃)). Then, let ⋃︀ 풞 := { 훿∈푑 푐훿 | 푑 ∈ 풟}. Evidently, 풞(휈, 휃) ≤ |풞| ≤ |풟| = 풟(휈, cf(휃)). Corollary 3.2. For every infinite cardinal 휃 and every cardinal 휆 ≥ 풟(휃, cf(휃)), we have 풞(휈, 휃) ≤ 휆 whenever 휈 ∈ Reg(휆) ∖ 휃. Proof. First, note that 풟(휃+푛, cf(휃)) ≤ max{휃+푛, 풟(휃, cf(휃))} for all 푛 < 휔. We now prove the contrapositive. Suppose that 휈 is a regular cardinal and 풞(휈, 휃) > 휆 > 휈 ≥ 휃. Then, by Lemma 3.1, 휈 = 휃+푛 for some 푛 < 3 and 풟(휈, cf(휃)) > 휆. It follows that 휆 < 풟(휈, cf(휃)) ≤ max{휈, 풟(휃, cf(휃))} ≤ max{휆, 풟(휃, cf(휃)}, and hence 풟(휃, cf(휃)) > 휆. Remark 3.3. See [Koj16] for a study of the map 휃 ↦→ 풟(휃, cf(휃)) over the class of singular cardinals. A main aspect of the upcoming proofs is the analysis of local versus global features of a function. For this, it is useful to establish the following lemma. 3Take note of the direction of the containment. 6 MAXWELL LEVINE AND ASSAF RINOT Lemma 3.4. For any ordinal 휁 of uncountable cofinality, there exists a class map 휓휁 : ORD → cf(휁) satisfying the following. For every ordinal 훼 with cf(훼) = cf(휁), every function 푓 : 훼 → ORD and every stationary 푠 ⊆ 훼 such that 푓 푠 is strictly increasing and converging to 휁, there exists a club 푐 ⊆ 훼 such that (휓휁 ∘ 푓) (푐 ∩ 푠) is strictly increasing. Proof. Let 휈 be some regular uncountable cardinal, and let 휁 be an ordinal of cofinality 휈. For every ordinal 훼 of cofinality 휈, fix a strictly increasing function 휋훼 : 휈 → 훼 whose image is a club in 훼. Define 휓휁 : ORD → 휈 by letting: {︃ min{푖 < 휈 | 휂 < 휋 (푖)}, if 휂 < 휁; 휓 (휂) := 휁 휁 0, otherwise. Now, suppose that we are given a function 푓 : 훼 → ORD along with a stationary 푠 ⊆ 훼 ¯ on which 푓 is strictly increasing and converging to 휁. Let 푓 := 휓휁 ∘ 푓 ∘ 휋훼, which is a function from 휈 to 휈. Consider the club푐 ¯ := {훿¯ < 휈 | 푓¯[훿¯] ⊆ 훿¯}, and the set 푡¯ := {훿¯ < 휈 | ¯ ¯ ¯ ¯ 휋훼(훿) ∈ 푠 & 푓(훿) < 훿}. If 푡¯ is stationary, then by Fodor’s lemma, we may fix some stationary 푡ˆ ⊆ 푡¯ and some ¯ ˆ ˆ 푖 < 휈 such that 푓[푡] = {푖}. As 휋훼[푡] is a cofinal (indeed, stationary) subset of 푠, and 푓 푠 is strictly increasing and converging to 휁, we may find a large enough 훿 ∈ 휋훼[푡ˆ] such that ¯ −1 ¯ ¯ ¯ 휂 := 푓(훿) is greater than 휋휁 (푖). But then, for 훿 := 휋훼 (훿), we have 푓(훿) = 휓휁 (푓(휋훼(훿))) = ¯ 휓휁 (푓(훿)) = 휓휁 (휂) > 푖, contradicting the fact that 훿 ∈ 푡ˆ. So 푡¯ is nonstationary, and hence we may find a club 푐 in 훼 with 푐 ⊆ 휋훼[¯푐 ∖ 푡¯]. To see that (휓휁 ∘ 푓) (푐 ∩ 푠) is strictly increasing, let 훾 < 훿 be an arbitrary pair of elements of 푐 ∩ 푠. −1 ¯ −1 ¯ ¯ ¯ Put훾 ¯ := 휋훼 (훾) and 훿 := 휋훼 (훿). As 훿 ∈ 푐¯ ∖ 푡 and 휋훼(훿) ∈ 푠, we indeed have ¯ ¯ ¯ (휓휁 ∘ 푓)(훾) = 푓(¯훾) < 훿 ≤ 푓(훿) = (휓휁 ∘ 푓)(훿). Theorem 3.5. Suppose: ∙ 휆 is a singular cardinal; ⃗ + ∙ 푓 = ⟨푓훽 | 훽 < 휆 ⟩ is a scale for 휆; ∙ 휈 is a regular uncountable cardinal ̸= cf(휆); ∙ 푆 and 푇 are subsets of 휆+; ⃗ 휆+ 4 ∙ 퐺(푓) ∩ Tr(푆) ∩ 퐸휈 ∩ 푇 is stationary. For every (finite or infinite) cardinal 휃 ≤ 휈, if any of the following holds true: (i) 휈 is an infinite successor cardinal and 풞(휈, 휃) ≤ 휆; (ii)2 휈 ≤ 휆, then Π(푆, 휃, 푇 ) holds. Proof. Let 휃 ≤ 휈 be cardinal satisfying Clause (i) or (ii). We shall prove that Π(푆, 휃, 푇 ) holds −1 by exhibiting a function ℎ : 푆 → 휃 and some stationary 푇0 ⊆ 푇 such that 푇0 ⊆ Tr(ℎ {휏}) for all 휏 < 휃. ⃗ 휆+ Denote 푇0 := 퐺(푓) ∩ Tr(푆) ∩ 퐸휈 ∩ 푇 . For every 푖 < cf(휆), denote 휆푖 := sup{푓훽(푖) | + ⃗ 훽 < 휆 }, so that 휆 := ⟨휆푖 | 푖 < cf(휆)⟩ is a strictly increasing sequence of regular cardinals, ⃗ converging to 휆. For each 훼 ∈ 푇0, since 훼 is a good point for 푓, we use Fact 2.3(2) to fix an 4Recall Definition 2.2. PARTITIONING A REFLECTING STATIONARY SET 7 ∏︀ ⃗ ⃗ exact upper bound 푒훼 ∈ 휆 for 푓 훼 such that cf(푒훼(푖)) = 휈 for all 푖 < cf(휆) with 휆푖 > 휈. Of course, we may also assume that 푒훼(푖) > 0 for all 푖 < cf(휆). Claim 3.5.1. Let 훼 ∈ 푇0. Then at least one of the following holds true: (1) there is an 푖 < cf(휆) such that 훿 ↦→ 푓훿(푖) is strictly increasing over some stationary subset of 푆 ∩ 훼. (2) there is an 푖 < cf(휆) with 휆푖 > 휈 such that, for cofinally many 훾 < 푒훼(푖), {훽 ∈ 푆 ∩훼 | 훾 ≤ 푓훽(푖) < 푒훼(푖)} is stationary in 훼. Proof. Suppose that Clause (1) fails. As cf(훼) = 휈 and 휈 ̸= cf(휆), this must mean that 휈 > cf(휆). Let 푘 < cf(휆) be the least to satisfy 휆푘 > 휈. Define 푔 : cf(휆) → 휆 by letting 푔(푖) := 0 for all 푖 < 푘, and letting 푔(푖) := sup{훾 < 푒훼(푖) | {훽 ∈ 푆 ∩ 훼 | 훾 ≤ 푓훽(푖) < 푒훼(푖)} is stationary in 훼} + 1 ∏︀ whenever 푘 ≤ 푖 < cf(휆). Towards a contradiction, suppose that 푔 ∈ 푖 For each 훼 ∈ 푇0, fix some 푖훼 as in the claim. Then fix some stationary 푇1 ⊆ 푇0 and some * * 푖 < cf(휆) such that 푖훼 = 푖 for all 훼 ∈ 푇1. Now, fix 휀 < 휆푖* and some stationary 푇2 ⊆ 푇1 * such that 푒훼(푖 ) = 휀 for all 훼 ∈ 푇2. Let 퐸 be some club in 휀 with otp(퐸) = cf(휀). Claim 3.5.2. Let 훼 ∈ 푇2. Then at least one of the following holds true: * (1) 훿 ↦→ 푓훿(푖 ) is strictly increasing over some stationary subset of 푆 ∩ 훼. (2) there is 퐷 ∈ [퐸]휈 such that, for any pair of ordinals 훾 < 훿 from 퐷, {훽 ∈ 푆 ∩ 훼 | * 훾 < 푓훽(푖 ) < 훿} is stationary in 훼. Proof. Suppose that Clause (1) fails. In particular, cf(휀) = 휈. So, to prove that Clause (2) holds, it suffices to show that, for all 훾 ∈ 퐸, there is a large enough 훿 ∈ 퐸 such that * {훽 ∈ 푆 ∩ 훼 | 훾 < 푓훽(푖 ) < 훿} is stationary in 훼. Thus, let 훾 ∈ 퐸 be arbitrary. Fix a strictly increasing function 휋0 : 휈 → 훼 whose image is a club in 훼, and a strictly * increasing function 휋1 : 휈 → 휀 whose image is 퐸. As 훼 ∈ 푇2, 푖 = 푖훼 and Clause (1) fails, we know that ¯ −1 * 푆 := 휋0 {훽 ∈ 푆 ∩ 훼 | 훾 + 1 ≤ 푓훽(푖 ) < 휀} is stationary in 휈. Define a function 휑 : 푆¯ → 휈 by stipulating: ¯ ¯ * ¯ 휑(훽) := min{훿 < 휈 | 푓휋0(훽¯)(푖 ) < 휋1(훿)}. Let 퐶ˆ := {훽¯ ∈ 푆¯ | 휑[훽¯] ⊆ 훽¯} and 푆ˆ := {훽¯ ∈ 푆¯ | 휑(훽¯) < 훽¯}. ˆ ˆ ˆ Note that if 푆 is nonstationary, then 푅 := 휋0[퐶 ∖ 푆] is a stationary subset of 푆 ∩ 훼, and for any pair of ordinals 훽 < 훽′ from 푅, we have −1 −1 ′ −1 ′ 휑(휋0 (훽)) < 휋0 (훽 ) ≤ 휑(휋0 (훽 )), 8 MAXWELL LEVINE AND ASSAF RINOT * −1 * meaning that 푓훽(푖 ) < 휋1(휑(휋0 (훽))) ≤ 푓훽′ (푖 ), and contradicting the fact that Clause (1) fails. So 푆ˆ must be stationary. ′ ˆ ¯ ¯ Fix a stationary subset 푆 ⊆ 푆 on which 휑 is constant, with value, say, 훿. Put 훿 := 휋1(훿), ′ ′ so that 훿 ∈ 퐸. Then 휋0[푆 ] is a stationary subset of 푆 ∩ 훼, and, for all 훽 ∈ 휋0[푆 ], we have * ¯ 훾 < 훾 + 1 ≤ 푓훽(푖 ) < 휋1(훿) = 훿, as sought. Let 푇3 denotes the set of all 훼 ∈ 푇2 for which Clause (2) of Claim 3.5.2 holds. Case 1. Suppose that 푇3 is stationary. For each 훼 ∈ 푇3, fix some 퐷훼 as in the claim. By replacing 퐷훼 with its closure, we may assume that 퐷훼 is a subclub of 퐸. As 퐸 is the order- 5 preserving continuous image of 휈 and as 풞(휈, 휃) ≤ 휆, we may fix some stationary 푇4 ⊆ 푇3 and some 퐷 ⊆ 퐸 of order-type 휃 such that 퐷 ⊆ 퐷훼 for all 훼 ∈ 푇4. Define ℎ : 푆 → 휃 by letting * * ℎ(훽) := 0 whenever 푓훽(푖 ) ≥ sup(퐷), and ℎ(훽) := sup(otp(푓훽(푖 )∩퐷)), otherwise. We claim −1 that 푇4 ⊆ Tr(ℎ {휏}) for all 휏 < 휃. To see this, let 훼 ∈ 푇4 and 휏 < 휃 be arbitrary. Let 훾 denote the unique element of 퐷 such that otp(퐷∩훾) = 휏. Let 훿 := min(퐷∖(훾+1)). As 훾 < 훿 ′ * is a pair of elements from 퐷훼, we know that 푆 := {훽 ∈ 푆 ∩ 훼 | 훾 < 푓훽(푖 ) < 훿} is stationary. ′ * Now, for each 훽 ∈ 푆 , we have ℎ(훽) = sup(otp(푓훽(푖 ) ∩ 퐷)) = sup(otp((훾 + 1) ∩ 퐷)) = sup(휏 + 1) = 휏.6 + * Case 2. Suppose that 푇3 is nonstationary. Define 푓 : 휆 → 휆푖* by letting 푓(훿) := 푓훿(푖 ) + for all 훿 < 휆 . As 푇3 is nonstationary, the set 푇5 of all 훼 ∈ 푇2 for which 훿 ↦→ 푓(훿) is injective over some stationary 푆훼 ⊆ 푆 ∩ 훼, is stationary. Fix 휁 ≤ 휆푖* and some stationary subset 푇6 ⊆ 푇5 such that, for all 훼 ∈ 푇6, sup(푓[푆훼]) = 휁. Let 휓휁 be given by Lemma 3.4, and then set 휙 := (휓휁 ∘푓) 푆. Then 휙 is a function from 푆 to 휈 with the property that, for all 훼 ∈ 푇6, there exists a stationary 푠훼 ⊆ 푆 ∩ 훼 on which 휙 is strictly increasing. 휈 휈 Case 2.1. Suppose that 2 ≤ 휆. For each 푔 ∈ 휃, we attach a function ℎ푔 : 푆 → 휃 by 휈 letting ℎ푔 := 푔 ∘ 휙. We claim that for every 훼 ∈ 푇6, there is some 푔훼 ∈ 휃 such that, for all −1 휏 < 휃, ℎ푔훼 {휏} ∩ 훼 is stationary in 훼. Indeed, given 훼 ∈ 푇6, we fix a stationary 푠훼 ⊆ 푆 ∩ 훼 on which 휙 is injective, then fix a partition ⟨푅휏 | 휏 < 휃⟩ of 푠훼 into stationary sets, and then pick 푔 : 휈 → 휃 such that, for all 휏 < 휃 and 훿 ∈ 푅휏 , 푔(휙(훿)) = 휏. Evidently, for all 휏 < 휃, −1 ℎ푔 {휏} covers the stationary set 푅휏 . 휈 휈 Now, as 2 ≤ 휆, fix some stationary 푇7 ⊆ 푇6 and some 푔 ∈ 휃 such that 푔훼 = 푔 for all −1 훼 ∈ 푇7. Then 푇7 ⊆ Tr(ℎ푔 {휏}) for all 휏 < 휃. 휈 + Case 2.2. Suppose that 2 > 휆, so that 휈 is a successor cardinal, say 휈 = 휒 . Let ⟨퐴휉,휂 | 휉 < 휈, 휂 < 휒⟩ be an Ulam matrix over 휈 [Ula30]. That is: ⋃︀ ∙ for all 휉 < 휈, |휈 ∖ 휂<휒 퐴휉,휂| ≤ 휒; ′ ∙ for all 휂 < 휒 and 휉 < 휉 < 휈, 퐴휉,휂 ∩ 퐴휉′,휂 = ∅. 휈 Claim 3.5.3. Let 훼 ∈ 푇6. There exist 휂 < 휒 and 푥 ∈ [휈] such that, for all 휉 ∈ 푥, −1 휙 [퐴휉,휂] ∩ 훼 is stationary in 훼. −1 Proof. Suppose not. Then, for all 휂 < 휒, the set 푥휂 := {휉 < 휈 | 휙 [퐴휉,휂]∩훼 is stationary in 훼} ⋃︀ has size ≤ 휒. So 푋 := 휂<휒 푥휂 has size ≤ 휒, and we may fix 휉 ∈ 휈 ∖ 푋. It follows that, for 5Recall that by Lemma 3.1, 풞(휈, 휃) ≤ 2휈 . 6Note that in this case, we did not need to assume that 휈 is a successor cardinal. PARTITIONING A REFLECTING STATIONARY SET 9 −1 −1 ⋃︀ all 휂 < 휒, 휙 [퐴휉,휂] ∩ 훼 is nonstationary in 훼. Consequently, 휙 [ 휂<휒 퐴휉,휂] ∩ 훼 is nonsta- ⋃︀ tionary in 훼. However, 휂<휒 퐴휉,휂 contains a tail of 휈, contradicting the fact that there exists a stationary 푠훼 ⊆ 푆 ∩ 훼 on which 휙 is strictly increasing and converging to 휈. For each 훼 ∈ 푇6, fix 휂훼 and 푥훼 as in the claim. As 풞(휈, 휃) ≤ 휆, fix some stationary 푇8 ⊆ 푇6 + along with 휂 < 휒 and 푥 ⊆ 휈 of order-type 휃 such that 휂훼 = 휂 and 푥 ⊆ acc (푥훼) for all 훼 ∈ 푇8. Let ℎ : 푆 → 휃 be any function satisfying ℎ(훿) := sup(otp(푥 ∩ 휉)) whenever 휙(훿) ∈ 퐴휉,휂. We −1 claim that 푇8 ⊆ Tr(ℎ {휏}) for all 휏 < 휃. To see this, let 훼 ∈ 푇8 and 휏 < 휃 be arbitrary. Let ′ ′ ′ 휉 denote the unique element of 푥 such that otp(푥 ∩ 휉 ) = 휏. Put 휉 := min(푥훼 ∖ (휉 + 1)). As + ′ ′ ′ 푥 ⊆ acc (푥훼), we know that [휉 , 휉) ∩ 푥 = {휉 }, so that otp(푥 ∩ 휉) = otp(푥 ∩ (휉 + 1)) = 휏 + 1. ′ −1 As 휂훼 = 휂 and 휉 ∈ 푥훼, the set 푆 := 휙 [퐴휉,휂] ∩ 훼 is a stationary subset of 푆 ∩ 훼. Now, for ′ each 훿 ∈ 푆 , we have 휙(훿) ∈ 퐴휉,휂, meaning that ℎ(훿) = sup(otp(푥 ∩ 휉)) = sup(휏 + 1) = 휏, as sought. Remarks 3.6. (1) It follows from Theorem 3.5 together with Lemma 3.1 and Fact 2.3 that for every singular cardinal 휆 there exists a partition of 휆+ into 휆 many reflecting stationary sets. (2) By appealing to a refinement of Fact 2.3(2), implicitly stated in [SV10, Footnote 5],7 we infer from Theorem 3.5 and Lemma 3.1 that for every singular cardinal 휆, every regular cardinal 휃 with cf(휆) < 휃 < 휆, every 푆 ⊆ 휆+, and every stationary 푇 ⊆ 휆+ + Tr(푆) ∩ 퐸휃+3 , Π(푆, 휃 , 푇 ) holds. We now prove a variation of Theorem 3.5 that does not require 휈 to be a successor cardinal. Theorem 3.7. Suppose: ∙ 휆 is a singular cardinal; ⃗ + ∙ 푓 = ⟨푓훽 | 훽 < 휆 ⟩ is a scale for 휆; ∙ 휒 < 휇 < 휈 are cardinals in Reg(휆) ∖ {cf(휆)}; 휆+ 휆+ ∙ 푆 ⊆ 퐸휇 and 푇 ⊆ 퐸휈 are sets; ∙ 퐺(푓⃗) ∩ Tr(푆) ∩ 푇 is stationary. For every cardinal 휃 ≤ 휈 satisfying 풞(휈, 휃) ≤ 휆,8 Π(푆, 휃, 푇 ) holds. ⃗ Proof. Set 푇0 := 퐺(푓) ∩ Tr(푆) ∩ 푇 . 훼 Claim 3.7.1. Let 훼 ∈ 푇0. There exist 푖훼 < cf(휆) and 푆훼 ⊆ 퐸휒 such that Tr(푆훼) ∩ 푆 is stationary in 훼, and ⟨푓훾(푖훼) | 훾 ∈ 푆훼⟩ is strictly increasing. Proof. As 훼 is good, let us fix a cofinal 퐴 ⊆ 훼 and 푖 < cf(훼) such that, for all 훿 < 훾 푖 + 훼 from 퐴, 푓훿 < 푓훾. Now, for every 훾 ∈ acc (퐴) ∩ 퐸휒 , since 휒 ̸= cf(휆), we may fix a cofinal 푖훾 푎훾 ⊆ 퐴 ∩ 훾 along with 푖훾 < cf(휆) such that, for all 훿 ∈ 푎훾, 푓훿 < 푓훾. By possibly increasing 푖훾 + 푖훾, we may also assume that 푓훾 < 푓min(퐴∖(훾+1)). Next, for every 훽 ∈ 푆 ∩ acc(acc (퐴)), since cf(훽) = 휇 and 휇 ̸= cf(휆), we may find some 푖훽 < cf(휆) along with a stationary + 훽 푆훽 ⊆ acc (퐴) ∩ 퐸휒 such that, for all 훾 ∈ 푆훽, 푖훾 = 푖훽. Then, since 휈 ̸= cf(휆), we may find a + stationary 퐵 ⊆ 푆 ∩ acc(acc (퐴)) and 푖훼 < cf(휆) such that, for all 훽 ∈ 퐵, max{푖훽, 푖} = 푖훼. 7See Lambie-Hanson’s answer in https://mathoverflow.net/questions/296225. 8Note that if 휈 is not a successor cardinal, then, by Lemma 3.1, 풞(휈, 휃) < 휆 for all 휃 < 휈. 10 MAXWELL LEVINE AND ASSAF RINOT ⋃︀ Put 푆훼 := {푆훽 | 훽 ∈ 퐵}. Trivially, Tr(푆훼) ∩ 푆 covers the stationary set 퐵. Now, let ′ 휀 < 훾 be an arbitrary pair of elements from 푆훼. Find 훽 ≤ 훽 from 퐵 such that 휀 ∈ 푆훽 and + 훾 ∈ 푆훽′ . Let 휖 := min(퐴 ∖ (휀 + 1)). Since 훾 ∈ 푆훽′ ⊆ acc (퐴), we have 휀 < 휖 < 훾. As sup(푎훾) = 훾, we may also fix 훿 ∈ 푎훾 above 휖, so that 휀 < 휖 < 훿 < 훾. By the choice of 푖휀 and 푖휀 푖훾 푖 푖훾, respectively, we have 푓휀 < 푓휖 and 푓훿 < 푓훾. As 휖, 훿 ∈ 퐴, we also have 푓휖 < 푓훿. But 푖훼 푖훼 푖훼 푖훼 = max{푖훽, 푖훽′ , 푖} = max{푖휀, 푖훾, 푖}, so that, altogether, 푓휀 < 푓휖 < 푓훿 < 푓훾. Thus, we have established that ⟨푓훾(푖훼) | 훾 ∈ 푆훼⟩ is strictly increasing. For each 훼 ∈ 푇0, fix 푖훼 and 푆훼 in the claim. Then find a stationary 푇1 ⊆ 푇0 along with * * * 푖 < cf(휆) and 휁 < 휆 such that, for all 훼 ∈ 푇1, 푖훼 = 푖 and ⟨푓훾(푖 ) | 훾 ∈ 푆훼⟩ converges to 휁. + * + Define 푓 : 휆 → 휆푖* by letting 푓(훾) := 푓훾(푖 ) for all 훾 < 휆 . Let 휓휁 be given by Lemma 3.4, and then put 휙 := 휓휁 ∘ 푓. For each 훼 ∈ 푇1, pick a club 퐶훼 ⊆ 훼 such that 휙 (퐶훼 ∩ 푆훼) is strictly increasing and converging to 휈. For every 훽 ∈ 푆, fix a strictly increasing function 휋훽 : 휇 → 훽 whose image is a club in 훽. For all 휉 < 휈 and 휂 < 휇, let 퐴휉,휂 := {훽 ∈ 푆 | 휙(휋훽(휂)) = 휉}. 휈 Claim 3.7.2. Let 훼 ∈ 푇1. There exist 휂 < 휇 and 푥 ∈ [휈] such that, for all 휉 ∈ 푥, 퐴휉,휂 ∩ 훼 is stationary in 훼. Proof. Suppose not. Then, for all 휂 < 휇, the set 푥휂 := {휉 < 휈 | 퐴휉,휂 ∩ 훼 is stationary in 훼} ⋃︀ has size < 휈. So 푋 := 휂<휇 푥휂 has size < 휈, and 휉 := sup(푋) is smaller than 휈. Pick 훾 ∈ 퐶훼 ∩ 푆훼 with 휙(훾) > 휉. Next, fix a strictly increasing function 휋훼 : 휈 → 훼 whose image is a club in 훼. Let ¯ ¯ ¯ 푔 := 휙 ∘ 휋훼, so that 푔 is a function from 휈 to 휈. Consider 퐷 := {훽 < 휈 | 푔[훽] ⊆ 훽} which is a club in 휈, and 퐵 := Tr(푆훼) ∩ 푆 ∩ acc(퐶훼 ∖ 훾) ∩ acc(휋훼[퐷]) which is a stationary subset of 훼. Let 훽 ∈ 퐵 be arbitrary. We have that 푆훼 ∩ 훽 is stationary in 훽 and Im(휋훽) ∩ (퐶훼 ∖ 훾) ∩ 휋훼[퐷] is a club in 훽, and hence we may find 휂 < 휇 such that 휋훽(휂) ∈ 푆훼 ∩ 퐶훼 ∩ Im(휋훼) ∖ (훾 + 1). As 휋훽(휂) > 훾 is a pair of ordinals of 퐶훼 ∩ 푆훼, we infer that 휙(휋훽(휂)) > 휙(훾) > 휉. In addition, 휋훽(휂) ∈ Im(휋훼) ∩ 훽 and 훽 ∈ 휋훼[퐷], so that −1 −1 푔(휋훼 (휋훽(휂))) < 휋훼 (훽). Thus, we have established that for every 훽 ∈ 퐵, there exist 휂훽 < 휇 such that 휉 < −1 휙(휋훽(휂훽)) < 휋훼 (훽). As 퐵 is stationary in 훼 and cf(훼) = 휈 > 휇, we may fix a stationary ′ * ¯ * 퐵 ⊆ 퐵 on which the function 훽 ↦→ 휂훽 is constant with value, say, 휂 . So 훽 ↦→ 휙(휋휋훼(훽¯)(휂 )) is −1 ′ ′′ ′ * regressive over 휋훼 [퐵 ], and hence we may find a stationary 퐵 ⊆ 퐵 on which 훽 ↦→ 휙(휋훽(휂 )) * ′′ is constant with value, say, 휉 . Then 퐴휉*,휂* ∩ 훼 covers the stationary set 퐵 , contradicting * the fact that 휉 > 휉 = sup(푋). For each 훼 ∈ 푇1, fix 휂훼 and 푥훼 as in the claim. As 풞(휈, 휃) ≤ 휆, fix some stationary 푇2 ⊆ 푇1 + along with 휂 < 휒 and 푥 ⊆ 휈 of order-type 휃 such that 휂훼 = 휂 and 푥 ⊆ acc (푥훼) for all 훼 ∈ 푇2. Let ℎ : 푆 → 휃 be any function satisfying ℎ(훿) := sup(otp(푥 ∩ 휉)) whenever 훿 ∈ 퐴휉,휂. We −1 claim that 푇2 ⊆ Tr(ℎ {휏}) for all 휏 < 휃. To see this, let 훼 ∈ 푇2 and 휏 < 휃 be arbitrary. Let ′ ′ ′ 휉 denote the unique element of 푥 such that otp(푥 ∩ 휉 ) = 휏. Put 휉 := min(푥훼 ∖ (휉 + 1)). As + ′ ′ ′ 푥 ⊆ acc (푥훼), we know that [휉 , 휉) ∩ 푥 = {휉 }, so that otp(푥 ∩ 휉) = otp(푥 ∩ (휉 + 1)) = 휏 + 1. ′ As 휂훼 = 휂 and 휉 ∈ 푥훼, the set 푆 := 퐴휉,휂 ∩ 훼 is a stationary subset of 푆 ∩ 훼. Now, for PARTITIONING A REFLECTING STATIONARY SET 11 ′ each 훿 ∈ 푆 , we have 훿 ∈ 퐴휉,휂, meaning that ℎ(훿) = sup(otp(푥 ∩ 휉)) = sup(휏 + 1) = 휏, as sought. Next, we address the case that 휆 is regular. 휆+ Theorem 3.8. Suppose that 휇 < 휃 < 휆 are infinite regular cardinals, and 푇 ⊆ 퐸휃 is 휆+ stationary. Then Π(퐸휇 , 휃, 푇 ) holds. Proof. By [She91, Lemma 4.4], there is a sequence ⟨Γ푗 | 푖 < 휆⟩ such that: ⋃︀ 휆+ ∙ 푖<휆 Γ푗 = 퐸<휆; 푗 ∙ for each 푗 < 휆, there is a sequence ⟨퐶훼 | 훼 ∈ Γ푗⟩ such that, for every limit ordinal 푗 푗 훼 ∈ Γ푗, 퐶훼 is a club in 훼 of order-type < 휆, and for each훼 ¯ ∈ acc(퐶훼), we have 푗 푗 훼¯ ∈ Γ푗 and 퐶훼¯ = 퐶훼 ∩ 훼¯. 휆+ Fix 푗 < 휆 such that 푇 ∩ 퐸휃 ∩ Γ푗 is stationary. Then find 휀 < 휆 and a stationary 휆+ 푗 푇0 ⊆ 푇 ∩ 퐸휃 ∩ Γ푗 such that, for all 훼 ∈ 푇0, otp(퐶훼) = 휀. By [BR19, Lemma 3.1], we may fix a function Φ : 풫(휆+) → 풫(휆+) satisfying that for every 훼 ∈ acc(휆+) and every club 푥 in 훼: ∙ Φ(푥) is a club in 훼; ∙ acc(Φ(푥)) ⊆ acc(푥); ∙ if훼 ¯ ∈ acc(Φ(푥)), then Φ(푥) ∩ 훼¯ = Φ(푥 ∩ 훼¯); ∙ if otp(푥) = 휀, then otp(Φ(푥)) = 휃. 푗 휃 −1 For each 훼 ∈ Γ푗, let 퐶훼 := Φ(퐶훼). Fix 푔 : 휃 → 휃 such that, for all 휏 < 휃, 퐸휇 ∩ 푔 {휏} is stationary in 휃. Define ℎ : 휆+ → 휃 as follows: {︃ 푔(otp(퐶 )), if 훿 ∈ Γ & otp(퐶 ) < 휃; ℎ(훿) := 훿 푗 훿 0, otherwise. Now, let 훼 ∈ 푇0 and 휏 < 휃 be arbitrary. As 퐶훼 ∩ 훿 = 퐶훿 for all 훿 ∈ acc(퐶훼), we get that 휆+ ⟨otp(퐶훿) | 훼 ∈ acc(퐶훼)⟩ is a club in 휃, and hence {훿 ∈ 퐸휇 ∩ acc(퐶훼) | 푔(otp(퐶훿)) = 휏} is stationary in 훼. Remark 3.9. The proof of the preceding makes clear that if 휇 < 휆 are infinite regular 휆+ 휆+ cardinals, 휆 holds, and 푇 ⊆ 퐸휆 is stationary, then Π(퐸휇 , 휆, 푇 ) holds. We are now ready to derive Theorem A. 휆+ Proof of Theorem A. (1) Suppose that 휆 is inaccessible, so that 휆 = ℵ휆. Trivially, ⟨퐸휇 | 휆+ 휆+ 휇 ∈ Reg(휆)⟩ witnesses Π(퐸<휆, 휆, 퐸휆 ). Likewise, for cofinally many 휃 < 휆 (e.g., 휃 휆 휆 singular with 휃 = ℵ휃), ⟨퐸휇 | 휇 ∈ Reg(휃)⟩ witnesses Π(퐸<휃, 휃, 휆). (2) By Theorem 3.8. ⃗ + (3) By Fact 2.3(1), we may let 푓 = ⟨푓훽 | 훽 < 휆 ⟩ be some scale for 휆. Let 휈 := 휃 so that 휆+ 휆+ 휈 is a regular cardinal ̸= cf(휆). Let 푆 := 퐸휇 , and 푇 := 퐸휈 , so that Tr(푆) ⊇ 푇 . By ⃗ 휆+ Fact 2.3(2), 퐺(푓) ∩ 퐸휈 is stationary. So, by Theorem 3.5, Π(푆, 휃, 푇 ) holds. (4) Let 푓⃗ be some scale for 휆. Let 휈 := min({휃+2, 휃+3} ∖ {cf(휆)}), so that 휈 is a successor cardinal ̸= cf(휆). By Lemma 3.1 and as 휃 < cf(휃)+ < 휈, we have 풞(휈, 휃) = 휈 < 휆. 12 MAXWELL LEVINE AND ASSAF RINOT 휆+ 휆+ ⃗ 휆+ Let 푆 := 퐸휇 and 푇 := 퐸휈 , so that Tr(푆) ⊇ 푇 . Then 퐺(푓) ∩ 퐸휈 is stationary, and so, by Theorem 3.5, Π(푆, 휃, 푇 ) holds. We conclude this section by establishing a corollary that was promised at the end of the previous section. Corollary 3.10. For every singular cardinal 휆 and every 휃 < 휆, there exists a partition ⟨푆푖 | + 휆+ ⋂︀ 푖 < 휃⟩ of 휆 such that sup{휈 < 휆 | 퐸휈 ∩ 푖<휃 Tr(푆푖) is stationary} = 휆. Proof. Let 퐴 be a cofinal subset of 휆 such that each 휇 ∈ 퐴 is a cardinal satisfying 휇 > 휇 휆+ 휆+ max{휃, cf(휆)}. For each 휇 ∈ 퐴, fix a sequence ⟨푆푖 | 푖 < 휃⟩ witnessing Π(퐸휇 , 휃, 퐸휇++ ). Then + ⋃︀휃 ⋃︀ 휇 a ⋃︀ 휇 + ⟨휆 ∖ 푖=1 휇∈퐴 푆푖 ⟩ ⟨ 휇∈퐴 푆푖 | 1 ≤ 푖 < 휃⟩ is a partition of 휆 as sought. 4. Theorem B We now introduce a weak consequence of the principle SNR(휅, 휈) from [CDS95]: − 휅 Definition 4.1. SNR (휅, 휈, 푇 ) asserts that for every stationary 푇0 ⊆ 푇 ∩ 퐸휈 , there exists a function 휙 : 휅 → 휈 such that, for stationarily many 훼 ∈ 푇0, for some club 푐 in 훼, 휙 푐 is strictly increasing. The relationship between SNR−(...) and Π(...) includes the following. Theorem 4.2. Suppose: ∙ 휈 < 휅 are regular uncountable cardinals; ∙ 푆 is subset of 휅; 휅 ∙ 푇 ⊆ Tr(푆) ∩ 퐸휈 is stationary; ∙ SNR−(휅, 휈, 푇 ) holds; ∙ 휃 ≤ 휈 is a cardinal satisfying 풞(휈, 휃) < 휅. If any of the following holds true: (1) 휈 is a successor cardinal; 휅 (2) 푆 ⊆ 퐸휇 for some regular uncountable 휇 < 휅; then Π(푆, 휃, 푇 ) holds. Proof. Fix a function 휙 : 휅 → 휈, a stationary 푇0 ⊆ 푇 , and a sequence ⃗푐 = ⟨푐훼 | 훼 ∈ 푇0⟩ such that, for all 훼 ∈ 푇0, 푐훼 is a club in 훼 (of order-type 휈) on which 휙 is strictly increasing. (1) The proof is similar to that of Theorem 3.5, so we only give a sketch. Suppose that + 휈 = 휒 is a successor cardinal. Let ⟨퐴휉,휂 | 휉 < 휈, 휂 < 휒⟩ be an Ulam matrix over 휈. For every 훼 ∈ 푇0, 푆 ∩ 푐훼 is a stationary subset of 푆 ∩ 훼 on which 휙 is injective. Consequently, 휈 and as made by clear by the proof of Claim 3.5.3, there are 휂훼 < 휒 and 푥훼 ∈ [휈] such that, −1 for all 휉 ∈ 푥훼, 휙 [퐴휉,휂훼 ] ∩ 푆 ∩ 훼 is stationary in 훼. As 풞(휈, 휃) < 휅, fix some stationary + 푇1 ⊆ 푇0 along with 휂 < 휒 and 푥 ⊆ 휈 of order-type 휃 such that 휂훼 = 휂 and 푥 ⊆ acc (푥훼) for all 훼 ∈ 푇1. Let ℎ : 푆 → 휃 be any function satisfying ℎ(훿) := sup(otp(푥 ∩ 휉)) whenever −1 휙(훿) ∈ 퐴휉,휂. Then 푇1 ⊆ Tr(ℎ {휏}) for all 휏 < 휃. (2) The proof is similar to that of Theorem 3.7. For every 훽 ∈ 푆, fix a strictly increasing function 휋훽 : 휇 → 훽 whose image is a club in 훽. For all 휉 < 휈 and 휂 < 휇, let 퐴휉,휈 := {훽 ∈ 푆 | PARTITIONING A REFLECTING STATIONARY SET 13 휙(휋훽(휂)) = 휉}. As made clear by the proof of Claim 3.7.2, for every 훼 ∈ 푇0, there exist 휈 9 휂훼 < 휇 and 푥훼 ∈ [휈] such that, for all 휉 ∈ 푥훼, 퐴휉,휂훼 ∩ 훼 is stationary in 훼. As 풞(휈, 휃) < 휅, fix some stationary 푇1 ⊆ 푇0 along with 휂 < 휇 and 푥 ⊆ 휈 of order-type 휃 such that 휂훼 = 휂 and + 푥 ⊆ acc (푥훼) for all 훼 ∈ 푇1. Let ℎ : 푆 → 휃 be any function satisfying ℎ(훿) := sup(otp(푥∩휉)) whenever 훿 ∈ 퐴휉,휂. The verification that ℎ witnesses Π(푆, 휃, 푇 ) is by now routine. Theorem 4.3. Suppose: ∙ 휈 < 휅 are infinite regular cardinals; ∙ 푆 is subset of 휅; 휅 ∙ 푇 ⊆ Tr(푆) ∩ 퐸휈 is stationary; ∙ SNR−(휅, 휈, 푇 ) holds; ∙ 2휈 < 휅. Then ∐︀(푆, 휈, 푇 ) holds. ⃗ Proof. Suppose 휃 ≤ 휈, 푆 = ⟨푆푖 | 푖 < 휃⟩ is a sequence of stationary subsets of 푆, and ⋂︀ − 푇0 ⊆ 푇 ∩ 푖<휃 Tr(푆푖) is stationary. By SNR (휅, 휈, 푇 ), fix a function 휙 : 휅 → 휈, a stationary 푇1 ⊆ 푇0 and a sequence ⟨푐훼 | 훼 ∈ 푇1⟩ such that, for every 훼 ∈ 푇1, 푐훼 is a club in 훼 (of order-type 휈) on which 휙 is injective. 휃 Claim 4.3.1. Let 훼 ∈ 푇1. Then there exists a function ℎ : 휈 → 휃 such that Im(ℎ) ∈ [휃] and, for all 푖 ∈ Im(ℎ), {훿 ∈ 푆푖 ∩ 훼 | ℎ(휙(훿)) = 푖} is stationary in 훼. ⋂︀ Proof. Let 휋 : 푐훼 → 휈 denote the unique order-preserving bijection. As 훼 ∈ 푇1 ⊆ 푖<휃 Tr(푆푖), we know that ⟨휋[푆푖] | 푖 < 휃⟩ is a sequence of stationary subsets of 휈, so by Lemma 1.3, we ′ fix a sequence ⟨푆푖 | 푖 ∈ 퐼⟩ of pairwise disjoint sets such that: ∙ 퐼 is a cofinal subset of 휃; ′ ∙ for each 푖 ∈ 퐼, 푆푖 ⊆ 휋[푆푖] is stationary. −1 ¯ ′ Now, as 휙 ∘ 휋 is injective, it easy to find ℎ : 휈 → 퐼 satisfying that, for all 푖 ∈ 퐼 and 훿 ∈ 푆푖, −1 ¯ ℎ(휙(휋 (훿))) = 푖. Clearly, any such ℎ is as sought. 휈 For each 훼 ∈ 푇1, fix a function ℎ훼 as in the claim. Then, as 2 < 휅, we may find a stationary 푇2 ⊆ 푇1 and some ℎ : 휈 → 휃 such that ℎ훼 = ℎ for all 훼 ∈ 푇2. Let 퐼 := Im(ℎ). For ′ ′ each 푖 ∈ 퐼, let 푆푖 := {훿 ∈ 푆푖 | ℎ(휙(훿)) = 푖}. Clearly, ⟨푆푖 | 푖 ∈ 퐼⟩ is a sequence as sought. Proposition 4.4. Suppose that 휈 < 휆 = 휆휈 < 휅 ≤ 2휆 are infinite cardinals with 휈, 휅 regular. Then: − 휅 (1) SNR (휅, 휈, 퐸휈 ) holds; ∐︀ 휅 (2) (휅, 휈, 퐸휈 ) holds. Proof. (1) By a standard application of the Engelking-Kar lowicz theorem. 휈 (2) By Clause (1) and Theorem 4.3, noticing that 2 ≤ 휆 < 휅. 휅 Proposition 4.5. Suppose that 휈 < 휅 are regular uncountable cardinals, 푇 ⊆ 퐸휈 , and ⋃︀ −1 there exists a function 푓 : 휅 → 휈 such that 푇 ∩ 푖<휈 Tr(푓 {푖}) is nonstationary. Then SNR−(휅, 휈, 푇 ) holds. 9 Note that if ⃗푐 is coherent in the sense that |{푐훼 ∩ 훽 | 훼 ∈ 푇0, 훽 ∈ acc(푐훼)}| ≤ 1 for all 훽 < 휅, then we can also handle the case 휇 = ℵ0. This complements the result mentioned in Remark 3.9. 14 MAXWELL LEVINE AND ASSAF RINOT −1 ⋃︀ Proof. Fix such a function 푓 : 휅 → 휈. Denote 푆푖 := 푓 {푖}. Let 훼 ∈ 푇 ∖ 푖<휈 Tr(푆푖) be 푖 arbitrary. For each 푖 < 휈, fix a club 푐훼 in 훼 disjoint from 푆푖. Let 휋 : 휈 → 훼 denote the −1 푖 inverse collapse of some club in 훼, and let 푐훼 := 휋[a푖<휈 휋 [푐훼]]. Then 푐훼 is a club in 훼, and, −1 푖 −1 for all 훽 ∈ 푐훼 and 푖 < 휋 (훽), we have 훽 ∈ 푐훼 so that 푓(훽) ̸= 푖. Consequently, 푓(훽) ≥ 휋 (훽) ′ for all 훽 ∈ 푐훼. Therefore, there exists a club 푐훼 ⊆ 푐훼 on which 푓 is strictly increasing. ⃗ + Proposition 4.6. Suppose 푓 = ⟨푓훽 | 훽 < 휆 ⟩ is a scale for a singular cardinal 휆, and − + 휆+ ⃗ 10 휈 ∈ Reg(휆) ∖ {ℵ0, cf(휆)}. Then SNR (휆 , 휈, 퐸휈 ∩ 푉 (푓)) holds. 휆+ ⃗ Proof. Let 푇0 be an arbitrary stationary subset of 퐸휈 ∩ 푉 (푓). For each 훼 ∈ 푇0, fix a club 푖훼 푐훼 ⊆ 훼 and some 푖훼 < cf(휆) such that for any pair 훿 < 훾 of ordinals of 푐훼, 푓훿 < 푓훾. Fix a stationary 푇1 ⊆ 푇0, and ordinals 푖 < cf(휆), 휁 < 휆 such that, for all 훼 ∈ 푇1, ⟨푓훿(푖) | 훿 ∈ 푐훼⟩ is + strictly increasing and converging to 휁. Let 휓휁 be given by Lemma 3.4. Define 휙 : 휆 → 휈 + by letting 휙(훿) := 휓휁 (푓훿(푖)) for all 훿 < 휆 . Clearly, for every 훼 ∈ 푇1, there exists a club ′ 푐훼 ⊆ 푐훼 on which 휙 is strictly increasing. Corollary 4.7. Suppose that 휈, 휆 are cardinals with ℵ0 < cf(휈) = 휈 < cf(휆). Then: − + 휆+ (1) SNR (휆 , 휈, 퐸휈 ) holds; 휈 ∐︀ + 휆+ (2) If 2 ≤ 휆, then (휆 , 휈, 퐸휈 ) holds. Proof. (1) If 휆 is singular, then by Fact 2.3(1), let us fix a scale 푓⃗ for 휆. As 휈 < cf(휆), we 휆+ ⃗ have 퐸휈 ⊆ 푉 (푓). Now, appeal to Proposition 4.6. 휆+ Next, suppose that 휆 is regular. Let 푇0 be an arbitrary stationary subset of 퐸휈 . As made clear by the proof of Theorem 3.8, there exists a sequence ⟨퐶훼 | 훼 ∈ Γ⟩ such that: ∙ Γ ⊆ acc(휆+); ∙ for all 훼 ∈ Γ, 퐶훼 is a club in 훼; ∙ for all 훼 ∈ Γ and훼 ¯ ∈ acc(퐶훼),훼 ¯ ∈ Γ and 퐶훼¯ = 퐶훼 ∩ 훼¯; ∙ 푇1 := {훼 ∈ Γ ∩ 푇0 | otp(퐶훼) = 휈} is stationary. Now, define 휙 : 휅 → 휈 by letting 휙(훼) := otp(퐶훼) whenever 훼 ∈ Γ and otp(퐶훼) < 휈; otherwise, let 휙(훼) := 0. Then ⟨퐶훼 | 훼 ∈ 푇1⟩ witnesses that 휙 is as sought. (2) By Clause (1) and Theorem 4.3. We are now ready to derive Theorem B. Proof of Theorem B. If 휆 is a singular cardinal admitting a very good scale, then by Propo- − + 휆+ sition 4.6, SNR (휆 , 휈, 퐸휈 ) holds. If 휆 is a regular cardinal, then by Corollary 4.7(1), − + 휆+ SNR (휆 , 휈, 퐸휈 ) holds. Now, appeal to Theorem 4.3. Acknowledgments The main results of this paper were presented by the second author at the Arctic Set Theory Workshop 4, Kilpisj¨arvi,January 2019. He thanks the organizers for the invitation. 10Recall Definition 2.2. PARTITIONING A REFLECTING STATIONARY SET 15 References [BR19] Ari Meir Brodsky and Assaf Rinot. Distributive Aronszajn trees. Fund. Math., 2019. accepted March 2018. http://doi.org/10.4064/fm542-4-2018. [CDS95] James Cummings, Mirna Dzamonja, and Saharon Shelah. A consistency result on weak reflection. 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