Maximal almost disjoint families of functions

Abstract

Abstract. We study maximal almost disjoint (MAD) families of functions in ω 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.

Convergence of measures in forcing extensions

“…In this section we present some combinatorial results implied by the preservation of the ground model set of reals as a non-meager subset of the reals in the extension. Typical examples of notions of forcing preserving the ground model reals nonmeager include Sacks, side-by-side products of Sacks, Miller, and Silver (see Raghavan [24,Section 5]). The property is preserved by countable support iterations ([24, Theorem 61]).…”

Projective mad families

“…The answer to Question 16 is ''no'' for the case of ω-mad families. This follows from Corollary 38 of [10] (it talks only about ω ω , but its proof works for [ω] ω as well). Indeed, suppose that b > ω 1 and A is a Σ 1 2 definable ω-mad family.…”

“…Combining Corollary 53 and Theorem 65 from [10], we conclude that the ground model ω-mad families of functions remain so in forcing extensions by countable support iterations of a wide family of posets including Sacks and Miller forcings. If A ∈ V is a Π 1 1 definable almost disjoint family whose Π 1 1 definition is provided by formula ϕ(x), then ϕ(x) defines an almost disjoint family in any extension V of V (this is a straightforward consequence of Shoenfield's Absoluteness Theorem).…”

“…Since the size of A is at least b > ω 1 , A must contain a perfect set. But this cannot happen for an ω-mad family by [10,Corollary 38].…”

“…It is not known whether in ZFC one can prove the existence of Σ 1 1 mad families of functions or of ω-mad families of functions; see [10].…”