Let $\kappa$ be a regular, uncountable cardinal. Let $S\subseteq \kappa$ be a closed and unbounded set. Suppose that we partition $S$ into $<\kappa$ pieces. Does one of those pieces contain a closed and unbounded set?
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- 1$\begingroup$ If, instead of ZFC, you assume ZF + Axiom of Determinacy, then the answer is yes for $\kappa=\omega_1$. $\endgroup$Lajos Soukup– Lajos Soukup2024-03-01 16:52:49 +00:00Commented Mar 1, 2024 at 16:52
- $\begingroup$ @LajosSoukup That is exactly the kind of thing I was looking for. Do you have a reference? $\endgroup$Pace Nielsen– Pace Nielsen2024-03-01 17:06:36 +00:00Commented Mar 1, 2024 at 17:06
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1 Answer
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1 Not necessarily. In fact, we can split $\kappa$ into $\kappa$-many disjoint stationary sets; this (and more) is due to Solovay, see "stationary splitting."
- 2$\begingroup$ No set in such a splitting can include a club, because the other sets in the splitting are stationary. $\endgroup$Andreas Blass– Andreas Blass2024-02-29 16:28:40 +00:00Commented Feb 29, 2024 at 16:28