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PARTITIONING a REFLECTING STATIONARY SET 1. Introduction A 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 S of a regular uncountable cardinal 휅, there exists a partition hSi j i < 휅i of S into stationary sets. The standard proof involves the analysis of a certain C-sequence over a stationary subset of S; such a sequence exists in ZFC (cf. [LS09]), but assuming the existence of better C-sequences, stronger partition theorems follow. For instance: + ∙ (folklore) If 휅 = 휆 and 휆 holds, then any stationary subset S of 휅 may be par- titioned into non-reflecting stationary sets hSi j i < 휅i. That is, for all i < 휅, Si is stationary, but Tr(Si) := f훽 < sup(Si) j cf(훽) > ! & Si \ 훽 is stationary in 훽g is empty. ∙ [Rin14, Lemma 3.2] If (휅) holds, then for every stationary S ⊆ 휅, there exists a ~ coherent C-sequence C = hC훼 j 훼 < 휅i such that Si := f훼 2 S j min(C훼) = ig is ~ stationary for all i < 휅. The coherence of C implies that the elements of hTr(Si) j i < 휅i 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 S is a subset of a regular uncountable cardinal 휅 for which T Tr(S) is stationary. Can S be split into sets hSi j i < 휃i in such a way that 푖<휃 Tr(Si) is stationary? And how large can 휃 be? Remark. Note that for any sequence hSi j i < 휃i of pairwise disjoint subsets of 휅, the T 휅 intersection 푖<휃 Tr(Si) is a subset of E≥휃. Therefore, the only cardinals 휃 of interest are the ones for which 휅 n 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(S; 휃; T ) asserts the existence of a partition hSi j i < 휃i of S such that T T \ 푖<휃 Tr(Si) is stationary. In Magidor's model [Mag82, x2], Π(S; @ ;T ) holds for any two stationary subsets S ⊆ E@2 1 @0 and T ⊆ E@2 , and it is also easy to provide consistent affirmative answers to Question 1.1 @1 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(E휇 ; 휃; E휃 ) holds; 휃 휆+ 휆+ (3) If 2 ≤ 휆 and 휃 6= cf(휆), then Π(E휇 ; 휃; E휃 ) holds; 휆+ 휆+ (4) If 휆 is singular, then Π(E휇 ; 휃; E≤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 Q ~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 hSi j i < 휃i is a sequence of stationary 0 subsets of a regular uncountable cardinal 휅, with 휃 ≤ 휅. Then there exists a sequence hSi j i 2 Ii of pairwise disjoint stationary sets such that: 0 ∙ Si ⊆ Si for every i 2 I; ∙ I 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. (S; 휈; T ) asserts that for every 휃 ≤ 휈, every sequence hSi j i < 휃i of 0 T 0 subsets of S and every stationary T ⊆ T \ 푖<휃 Tr(Si), there exists a sequence hSi j i 2 Ii of pairwise disjoint stationary sets such that: 0 ∙ Si ⊆ Si for every i 2 I; 1Note that if we demand that I be equal to 휃, then we do not get a theorem in ZFC. PARTITIONING A REFLECTING STATIONARY SET 3 ∙ I is a cofinal subset of 휃; 0 T 0 ∙ T \ i2I Tr(Si) is stationary. It is proved: Theorem B. Suppose that 휈 < 휆 are uncountable cardinals with 휈 6= cf(휆), and 2휈 ≤ 휆. ` + 휆+ Then, any of the following implies that (휆 ; 휈; E휈 ) 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 E휃 := f훼 < 휅 j cf(훼) = 휃g; E6=휃, 휅 휅 휅 + E>휃, E≥휃 and E≤휃 are defined similarly. For a set of ordinals a, we write acc (a) := f훼 < sup(a) j sup(a \ 훼) = 훼 > 0g and acc(a) := a \ acc+(a). The class of all ordinals is denoted by ORD. We also let Reg(휅) := f휃 < 휅 j @0 ≤ cf(휃) = 휃g. 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 = h휆i j i < cf(휆)i is a strictly increasing sequence of regular cardinals, converging to 휆. For any two functions f; g 2 Q ~휆 and i < cf(휆), we write f <i g iff, for all j 2 cf(휆) n i, f(j) < g(j). We also write f <* g to expresses that f <i g for some i < cf(휆). ~ + Definition 2.2. Suppose that 휆 is a singular cardinal. A sequence f = hf훾 j 훾 < 휆 i is said to be a scale for 휆 iff there exists a sequence ~휆 as in the previous definition, such that: + Q ~ ∙ for every 훾 < 휆 , f훾 2 휆; + * ∙ for every 훾 < 훿 < 휆 , f훾 < f훿; Q ~ + * ∙ for every g 2 휆, there exists 훾 < 휆 such that g < f훾. An ordinal 훼 < 휆+ is said to be: ∙ good with respect to f~ iff there exist a cofinal subset A ⊆ 훼 and i < cf(휆) such that, i for every pair of ordinals 훾 < 훿 from A, f훾 < f훿; ∙ very good with respect to f~ iff there exist a club C ⊆ 훼 and i < cf(휆) such that, for i every pair of ordinals 훾 < 훿 from C, f훾 < f훿. We also let: ~ 휆+ ~ ∙ G(f) := f훼 2 E6=cf(휆) j 훼 is good with respect to fg; ~ 휆+ ~ ∙ V (f) := f훼 2 E6=cf(휆) j 훼 is very good with respect to fg. 휆+ ~ ~ ~ Clearly, E<cf(휆) ⊆ V (f) ⊆ G(f). It is also not hard to see that if f;~g are two scales in the same product Q ~휆, then they interleave each other on a club, so that G(f~) 4 G(~g) is nonstationary. This means that, up to a club, the set of good points is in fact an invariant 4 MAXWELL LEVINE AND ASSAF RINOT of ~휆. We shall soon discuss the question of whether the same is true for the set of very good points, but let us first recall a few fundamental results of Shelah.2 Fact 2.3 (Shelah, [She93],[She94]). For every singular cardinal 휆: (1) There exists a scale for 휆; ~ ~ 휆+ (2) For any scale f for 휆 and every 휃 2 Reg(휆) n fcf(휆)g, the intersection G(f) \ E휃 is stationary; ~ + Q 휆+ (3) If f = hf훾 j 훾 < 휆 i is a scale for 휆 in a product i<cf(휆) 휆i and 훼 2 E6=cf(휆), ~ Q then 훼 2 G(f) iff there exists e 2 i<cf(휆) 휆i satisfying cf(e(i)) = cf(훼) whenever ~ 휆i > cf(훼), and such that e forms an exact upper bound for f 훼, i.e.: * ∙ for all 훾 < 훼, f훾 < e; Q * * ∙ for all g 2 i<cf(휆) 휆i with g < e, there is 훾 < 훼 with g < f훾.
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