Invariance Properties of Almost Disjoint Families

We answer a question of Garcia-Ferreira and Hrušák by consistently constructing a MAD family maximal in the Katětov order. We also answer several questions of Garcia-Ferreira.

On Weakly Tight Families

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when 𝔠 < ℵω, we construct a weakly tight family under the hypothesis 𝔰 ≤ 𝔟 < ℵω. The case when 𝔰 < 𝔟 is handled in ZFC and does not require 𝔟 < ℵω, while an additional PCF type hypothesis, which holds when 𝔟 < ℵω is used to treat the case 𝔰 = 𝔟. The notion of a weakly tight family is a natural weakening of the well-studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hrušák and García Ferreira [8], who applied it to the Katétov order on almost disjoint families.