Maximal almost disjoint families, determinacy, and forcing

TL;DR

This paper investigates the existence of maximal almost disjoint families relative to Borel ideals, showing under various determinacy axioms that such families do not exist for certain classes of ideals, extending classical results.

Contribution

It proves the nonexistence of analytic and projective $ ext{MAD}$ families for specific Borel ideals under determinacy axioms, combining descriptive set theory and forcing techniques.

Findings

No analytic infinite $ ext{MAD}$ families for $F_\sigma$ ideals.

No projective $ ext{MAD}$ families under Projective Determinacy.

No infinite $ ext{MAD}$ families under full Axiom of Determinacy + $V= ext{L}( extbf{R})$.

Abstract

We study the notion of $\mathcal J$-MAD families where $\mathcal J$ is a Borel ideal on $\omega$. We show that if $\mathcal J$ is an arbitrary $F_\sigma$ ideal, or is any finite or countably iterated Fubini product of $F_\sigma$ ideals, then there are no analytic infinite $\mathcal J$-MAD families, and assuming Projective Determinacy there are no infinite projective $\mathcal J$-MAD families; and under the full Axiom of Determinacy + $V=\mathbf{L}(\mathbb{R})$ there are no infinite $\mathcal J$-mad families. These results apply in particular when $\mathcal J$ is the ideal of finite sets $\mathrm{Fin}$, which corresponds to the classical notion of MAD families. The proofs combine ideas from invariant descriptive set theory and forcing.