Combinatorial properties of MAD families

Abstract

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

43 pages. Submitted. arXiv admin note: text overlap with arXiv:1810.09680