Almost disjoint families under determinacy

Abstract

For each cardinal κ, let B(κ) be the ideal of bounded subsets of κ and Pκ(κ) 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(κ) almost disjoint family. Also, the paper will completely characterize for which cardinals κ there is a nontrivial maximal Pκ(κ) almost disjoint family when κ is not an uncountable cardinal of countable cofinality. More precisely, the following will be shown. Assuming AD+, for all κ<Θ, there are no maximal B(κ) almost disjoint families A such that ¬(|A|<cof(κ)). For all κ<Θ, if cof(κ)>ω, then there are no maximal Pκ(κ) almost disjoint families A so that ¬(|A|<cof(κ)). Assume AD and V=L(R) (or more generally, AD+ and V=L(P(R))). For any cardinal κ, there is a maximal B(κ) almost disjoint family A so that ¬(|A|<cof(κ)) if and only if cof(κ)≥Θ. For any cardinal κ with cof(κ)>ω, there is a maximal Pκ(κ) almost disjoint family if and only if cof(κ)≥Θ.