Almost disjoint families under determinacy. (English) Zbl 07794560
Summary: For each cardinal \(\kappa \), let \(\mathcal{B}(\kappa)\) be the ideal of bounded subsets of \(\kappa\) and \(\mathcal{P}_\kappa(\kappa)\) be the ideal of subsets of \(\kappa\) of cardinality less than \(\kappa \). Under determinacy hypothesis, this paper will completely characterize for which cardinals \(\kappa\) there is a nontrivial maximal \(\mathcal{B}(\kappa)\) almost disjoint family. Also, the paper will completely characterize for which cardinals \(\kappa\) there is a nontrivial maximal \(\mathcal{P}_\kappa(\kappa)\) almost disjoint family when \(\kappa\) is not an uncountable cardinal of countable cofinality. More precisely, the following will be shown. Assuming \(\mathsf{AD}^+\), for all \(\kappa < \Theta \), there are no maximal \(\mathcal{B}(\kappa)\) almost disjoint families \(\mathcal{A}\) such that \(\neg(| \mathcal{A} | < \operatorname{cof}(\kappa))\). For all \(\kappa < \Theta \), if \(\operatorname{cof}(\kappa) > \omega \), then there are no maximal \(\mathcal{P}_\kappa(\kappa)\) almost disjoint families \(\mathcal{A}\) so that \(\neg(| \mathcal{A} | < \operatorname{cof}(\kappa))\). Assume \(\mathsf{AD}\) and \(V = L(\mathbb{R})\) (or more generally, \( \mathsf{AD}^+\) and \(V = L(\mathcal{P}(\mathbb{R}))\)). For any cardinal \(\kappa \), there is a maximal \(\mathcal{B}(\kappa)\) almost disjoint family \(\mathcal{A}\) so that \(\neg(| \mathcal{A} | < \operatorname{cof}(\kappa))\) if and only if \(\operatorname{cof}(\kappa) \geq \Theta \). For any cardinal \(\kappa\) with \(\operatorname{cof}(\kappa) > \omega \), there is a maximal \(\mathcal{P}_\kappa(\kappa)\) almost disjoint family if and only if \(\operatorname{cof}(\kappa) \geq \Theta\).
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