New examples of MAD families with pseudocompact hyperspaces

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 $p$.