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. 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 $¬(|A| < cof(κ))$.
This is joint work with W. Chan and S. Jackson. Our work is inspired by work of Schrittesser and Törnquist, and of Neeman and Norwood that showed there are no maximal almost disjoint families on ω under $AD^+$.