Fin-intersecting MAD families

TL;DR

This paper introduces fin-intersecting MAD families, exploring their existence under various set-theoretic assumptions and their relation to pseudocompactness in Isbell-Mrówka spaces.

Contribution

It defines fin-intersecting MAD families and demonstrates their existence under certain set-theoretic conditions, expanding the understanding of pseudocompact MAD families.

Findings

Existence of fin-intersecting MAD families under =

Existence under < in ZFC

Persistence of fin-intersecting MAD families after Cohen and Random reals

Abstract

We introduce a new class of almost disjoint families which we call fin-intersecting almost disjoint families. They are related to almost disjoint families whose Vietoris Hyperspace of their Isbell-Mr\'owka spaces are pseudocompact. We show that under $\mathfrak p=\mathfrak c$ fin-intersecting MAD families exist generically and they also exist if $\mathfrak{a<s}$, but that there are also non fin-intersecting MAD families in ZFC. We also show that under CH, there exists fin intersecting MAD families which remain like that after adding an arbitrary quantity of Cohen reals and Random reals. These results give more models in which pseudocompact MAD families exist.