Combinatorial properties of MAD families

TL;DR

This paper investigates the combinatorial properties of MAD families, characterizing Shelah-Stepr ext={a}ns ideals, and demonstrating their indestructibility and consistency results within set theory.

Contribution

It characterizes Shelah-Stepr ext={a}ns ideals among Borel ideals, proves their indestructibility properties, and establishes the consistency of their non-existence under certain set-theoretic assumptions.

Findings

A Borel ideal is Shelah-Stepr ext={a}ns iff it is Kat ext={e}tov above fin×fin.

Shelah-Stepr ext={a}ns MAD families are Cohen and random indestructible.

It is consistent that non( ext{mathcal{M}})= ext{ extonehalf}1 and no Shelah-Stepr ext={a}ns families of size ext{ extonehalf}1.

Abstract

We study some strong combinatorial properties of $\textsf{MAD}$ families. An ideal $\mathcal{I}$ is Shelah-Stepr\={a}ns if for every set $X\subseteq{\left[ \omega\right]}^{<\omega}$ there is an element of $\mathcal{I}$ that either intersects every set in $X$ or contains infinitely many members of it. We prove that a Borel ideal is Shelah-Stepr\={a}ns if and only if it is Kat\v{e}tov above the ideal $\textsf{fin}\times\textsf{fin}$. We prove that Shelah-Stepr\={a}ns $\textsf{MAD}$ families have strong indestructibility properties (in particular, they are both Cohen and random indestructible). We also consider some other strong combinatorial properties of $\textsf{MAD}$ families. Finally, it is proved that it is consistent to have $\mathrm{non}(\mathcal{M}) = {\aleph}_{1}$ and no Shelah-Stepr\={a}ns families of size ${\aleph}_{1}$.