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 SoukupCommented Mar 1 at 16:52
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$\begingroup$ @LajosSoukup That is exactly the kind of thing I was looking for. Do you have a reference? $\endgroup$– Pace NielsenCommented Mar 1 at 17:06
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1 Answer
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Not necessarily. In fact, we can split $\kappa$ into $\kappa$-many disjoint stationary sets; this (and more) is due to Solovay, see "stationary splitting."
answered Feb 29 at 16:07
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2$\begingroup$ No set in such a splitting can include a club, because the other sets in the splitting are stationary. $\endgroup$ Commented Feb 29 at 16:28