## 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:

 300000 Other combinatorial set theory 3e+55 Large cardinals
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