Compact Sets of Baire Class One Functions and Maximal Almost Disjoint Families

TL;DR

This paper proves that certain maximal almost disjoint families cannot exist in specific set-theoretic models using properties of Baire class one functions, extending classical dichotomy results to broader contexts.

Contribution

It introduces a novel application of compact Baire class one functions to set theory, providing new proofs and extensions of results about almost disjoint families in Solovay's model.

Findings

Analytic almost disjoint families are not maximal.

No maximal almost disjoint families exist in Solovay's model.

Extensions of classical dichotomy results to general pointwise bounded functions.

Abstract

We provide a proof that analytic almost disjoint families of infinite sets of integers cannot be maximal using a result of Bourgain about compact sets of Baire class one functions. Inspired by this and related ideas, we then provide a new proof of that there are no maximal almost disjoint families in Solovay's model. We then use the ideas behind this proof to provide an extension of a dichotomy result by Rosenthal and by Bourgain, Fremlin and Talagrand to general pointwise bounded functions in Solovay's model. We then show that the same conclusions can be drawn about the model obtained when we add a generic selective ultrafilter to the Solovay model.