Almost disjoint families under determinacy

For each cardinal kappa, let B(kappa) be the ideal of bounded subsets of kappa and 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 B(kappa) almost disjoint family. Also, the paper will completely characterize for which cardinals kappa there is a nontrivial maximal P-kappa(kappa) almost disjoint family when kappa is not an uncountable cardinal of countable cofinality. More precisely, the following will be shown.Assuming AD(+), for all kappa < Theta, there are no maximal B(kappa) almost disjoint families A such that <not sign>(|A| < cof(kappa)). For all kappa < Theta, if cof(kappa) > omega, then there are no maximal P-kappa(kappa) almost disjoint families A so that <not sign>(|A|<cof(kappa)).Assume AD and V = L(R) (or more generally, AD(+) and V = L(P(R))). For any cardinal kappa, there is a maximal B(kappa) almost disjoint family A so that <not sign>(|A|<cof(kappa)) if and only if cof(kappa) >= Theta. For any cardinal kappa with cof(kappa) > omega, there is a maximal P-kappa(kappa) almost disjoint family if and only if cof(kappa) >= Theta.(c) 2023 Elsevier Inc. All rights reserved.