Almost disjoint families under determinacy.

For each cardinal κ , let B (κ) be the ideal of bounded subsets of κ and P κ (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 P κ (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 P κ (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 P κ (κ) almost disjoint family if and only if cof (κ) ≥ Θ. [ABSTRACT FROM AUTHOR]