Combinatorial properties of MAD families

We study some strong combinatorial properties of $\mathsf{\text{MAD}}$ families. An ideal $\mathcal{I}$ is Shelah-Steprāns 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-Steprāns if and only if it is Katětov above the ideal $\mathsf{\text{fin}}\times \mathsf{\text{fin}}$. We prove that Shelah-Steprāns $\mathsf{\text{MAD}}$ families have strong indestructibility properties (in particular, they are both Cohen and random indestructible). We also consider some other strong combinatorial properties of $\mathsf{\text{MAD}}$ families. Finally, it is proved that it is consistent to have $\mathrm{n}\mathrm{o}\mathrm{n}(\mathcal{M})={\mathrm{\aleph }}_1$ and no Shelah-Steprāns families of size ${\mathrm{\aleph }}_1$.