Maximal almost disjoint families of functions

Abstract

We study maximal almost disjoint (MAD) families of functions in ω that satisfy certain strong combinatorial properties. In particular, we study the notions of strongly and very MAD families of functions. We introduce and study a hierarchy of combinatorial properties lying between strong MADness and very MADness. Proving a conjecture of Brendle, we show that if cov(M) < ae, then there no very MAD families. We answer a question of Kastermans by constructing a strongly MAD family from b = c. Next, we study the indestructibility properties of strongly MAD families, and prove that the strong MADness of strongly MAD families is preserved by a large class of posets that do not make the ground model reals meager. We solve a well-known problem of Kellner and Shelah by showing that a countable support iteration of proper posets of limit length does not make the ground model reals meager if no initial segment does. Finally, we prove that the weak Freese–Nation property of P(ω) implies that all strongly MAD families have size at most א1.