Combinatorial properties of MAD families

We study some strong combinatorial properties of $\textsf{MAD}$ families. An ideal $\mathcal{I}$ is Shelah-Steprāns if for every set $X\subseteq{\left[ ω\right]}^{<ω}$ 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-Steprāns if and only if it is Katětov above the ideal $\textsf{fin}\times\textsf{fin}$. We prove that Shelah-Steprāns $\textsf{MAD}$ 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-Steprāns families of size ${\aleph}_{1}$.

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