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 (|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 (|A|(|A|= 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.

Automated summary

Under the determinacy hypothesis, this paper completely characterizes the existence of nontrivial maximal almost disjoint families for specific cardinals kappa, considering the ideals of bounded subsets and subsets of cardinality less than kappa.