New examples of MAD families with pseudocompact hyperspaces

TL;DR

This paper explores the existence of MAD families with pseudocompact hyperspaces under certain set-theoretic assumptions, introduces new examples of non-fin-intersecting MAD families, and analyzes their structural properties.

Contribution

It demonstrates that specific set-theoretic principles imply the existence of MAD families with pseudocompact Vietoris hyperspaces and provides new examples and properties of MAD families.

Findings

Set-theoretic assumptions imply MAD families with pseudocompact hyperspaces.

New examples of non-fin-intersecting MAD families with special properties.

Characterization of the Baire number of  in relation to MAD families.

Abstract

We show that both $\mathfrak{ap=c}$ and $\diamondsuit(\mathfrak{b})$ imply the existence of MAD families with pseudocompact Vietoris hyperspace, substantially expanding the list of models where their existence is known. We also discuss some properties on the structure of fin-intersecting MAD families by providing some new examples of non-fin-intersecting MAD families with special properties. We show that the Baire number of $\omega^*$ is strictly larger than $\mathfrak c$ if and only if for every MAD family $\mathcal A$, $\Psi(\mathcal A)$ and its Vietoris hyperspace are $p$-pseudocompact for some free ultrafilter $p$.