Almost disjoint families under determinacy

Abstract

For each cardinal κ, let $B\left(\kappa \right)$ be the ideal of bounded subsets of κ and $P_{\kappa }\left(\kappa \right)$ be the ideal of subsets of κ of cardinality less than κ. Under determinacy hypothesis, this paper will completely characterize for which cardinals κ there is a nontrivial maximal $B\left(\kappa \right)$ almost disjoint family. Also, the paper will completely characterize for which cardinals κ there is a nontrivial maximal $P_{\kappa }\left(\kappa \right)$ almost disjoint family when κ is not an uncountable cardinal of countable cofinality. More precisely, the following will be shown.

Assuming ${\mathrm{AD}}^{+}$, for all $\kappa <\Theta$, there are no maximal $B\left(\kappa \right)$ almost disjoint families $A$ such that $\neg \left(\right|A|<\mathrm{cof}(\kappa \left)\right)$. For all $\kappa <\Theta$, if $\mathrm{cof}\left(\kappa \right)>\omega$, then there are no maximal $P_{\kappa }\left(\kappa \right)$ almost disjoint families $A$ so that $\neg \left(\right|A|<\mathrm{cof}(\kappa \left)\right)$.

Assume $\mathrm{AD}$ and $V=L\left(R\right)$ (or more generally, ${\mathrm{AD}}^{+}$ and $V=L\left(P\right(R\left)\right)$). For any cardinal κ, there is a maximal $B\left(\kappa \right)$ almost disjoint family $A$ so that $\neg \left(\right|A|<\mathrm{cof}(\kappa \left)\right)$ if and only if $\mathrm{cof}\left(\kappa \right)\ge \Theta$. For any cardinal κ with $\mathrm{cof}\left(\kappa \right)>\omega$, there is a maximal $P_{\kappa }\left(\kappa \right)$ almost disjoint family if and only if $\mathrm{cof}\left(\kappa \right)\ge \Theta$.