Combinatorial properties of MAD families

Abstract We study some strong combinatorial properties of 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 fin $\times $ fin , and that Shelah–Steprāns MAD families have strong indestructibility properties (in particular, they are both Cohen and random indestructible). We also consider some other strong combinatorial properties of MAD families. It is proved that it is consistent to have $\operatorname {\mathrm {non}}(\mathcal {M}) = {\aleph }_{1}$ and no Shelah–Steprāns families of size ${\aleph }_{1}$ . We develop a general machinery for producing models of set theory with $\operatorname {\mathrm {non}}(\mathcal {M}) = {\aleph }_{1}$ in order to prove this result.