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Splitting stationary sets in \(\mathcal P(\lambda )\). (English) Zbl 1250.03077

Summary: Let \(A\) be a non-empty set. A set \(S\subseteq \mathcal {P}(A)\) is said to be stationary in \(\mathcal{P}(A)\) if for every \(f: [A]^{<\omega } \rightarrow A\) there exists \(x\in S\) such that \(x\neq A\) and \(f``[x]^{<\omega } \subseteq x\). In this paper we prove the following: For an uncountable cardinal \(\lambda\) and a stationary set \(S\) in \(\mathcal{P}(\lambda)\), if there is a regular uncountable cardinal \(\kappa\leq \lambda\) such that \(\{x \in S: x \cap \kappa \in \kappa\}\) is stationary, then \(S\) can be split into \(\kappa\) disjoint stationary subsets.

MSC:

03E05 Other combinatorial set theory
03E55 Large cardinals
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References:

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