On weakly tight families

TL;DR

This paper constructs weakly tight families under certain set-theoretic hypotheses, extending the understanding of maximal almost disjoint families and their properties in different models.

Contribution

It introduces the construction of weakly tight families under hypotheses involving the splitting number and bounding number, generalizing previous results on maximal almost disjoint families.

Findings

Constructs weakly tight families assuming    < \u2205_\u03a9.

Handles the case  <  in ZFC without additional hypotheses.

Uses PCF theory to treat the case  = .

Abstract

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $\c < {\aleph}_{\omega}$, we construct a weakly tight family under the hypothesis $\s \leq \b < {\aleph}_{\omega}$. The case when $\s < \b$ is handled in $\ZFC$ and does not require $\b < {\aleph}_{\omega}$, while an additional PCF type hypothesis, which holds when $\b < {\aleph}_{\omega}$ is used to treat the case $\s = \b$. The notion of a weakly tight family is a natural weakening of the well studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hru{\v{s}}{\'a}k and Garc{\'{\i}}a Ferreira \cite{Hr1}, who applied it to the Kat\'etov order on almost disjoint families.