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On club-like principles on regular cardinals above \(\beth_\omega\). (English) Zbl 1022.03025

The main result of the paper reads as follows: Let \(\mu\) be a singular strong limit cardinal, and \(\lambda>\mu\) be a regular cardinal. Then for every large enough regular uncountable cardinal \(\kappa<\mu\) such that \(\lambda^{< \kappa}=\lambda\), one can find \(X_\delta\in P_\lambda(P_{\kappa^+} (\delta))\) for \(\delta< \lambda\) so that for each unbounded subset \(A\) of \(\lambda\), the set of all \(\delta< \lambda\) such that \(\text{cf} (\delta)= \kappa\) and \(\{a\in X_\delta \cap P(A):\sup(a)=\delta\}\neq\emptyset\) is stationary in \(\lambda\).
Reviewer’s comment: The result can be improved as follows. Let \(\mu\) be a singular strong limit cardinal, and \(\lambda>\mu\) be a cardinal with \(\text{cf} (\lambda)> \text{cf}(\mu)\). By a result of Shelah mentioned in the paper, given a cardinal \(\nu\geq\mu\), we have \(\text{cov} (\nu, \tau^+, \tau^+,\tau) =\nu\) for every large enough regular uncountable cardinal \(\tau< \mu\). Using an argument given in the paper, it follows that if \(\kappa< \mu\) is a large enough regular uncountable cardinal, then we can find \(X_\delta \in P_\lambda(P_{\kappa^+} (\delta))\) for \(\mu\leq\delta <\lambda\) so that \(P_{ \kappa^+} (\delta)= \{\bigcup e:e\in P_\kappa (X_\delta)\}\). Now given an unbounded subset \(A\) of \(\lambda\), there is a closed unbounded subset \(C\) of \(\lambda\) such that \(A\cap \delta\) is unbounded in \(\delta\) for every \(\delta\in C\). It is simple to see that \(\{a\in X_\delta\cap P(A):\sup(a) =\delta\}\neq \emptyset\) for every \(\delta\in C\) such that \(\mu\leq\delta\) and \(\text{cf} (\delta) = \kappa\).
Reviewer: P.Matet (Caen)

MSC:

03E05 Other combinatorial set theory
03E55 Large cardinals
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