Covering properties of $\omega$-mad families

TL;DR

This paper explores the properties of certain maximal almost disjoint families under different set-theoretic assumptions, demonstrating their indestructibility and impact on dominating reals.

Contribution

It shows that CH implies the existence of Cohen-indestructible mad families with specific forcing properties, and that certain equalities of cardinal invariants are consistent with their negation.

Findings

Existence of Cohen-indestructible mad families under CH

Mathias forcing adds dominating reals in this context

Consistency of = with the negation of this property

Abstract

We prove that CH implies the existence of a Cohen-indestructible mad family such that the Mathias forcing associated to its filter adds dominating reals, while $\mathfrak b=\mathfrak c$ is consistent with the negation of this statement as witnessed by the Laver model for the consistency of Borel's conjecture.