On weakly tight families

Abstract

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $\c < {\aleph}_ω$, we construct a weakly tight family under the hypothesis $\s \leq \b < {\aleph}_ω$. The case when $\s < \b$ is handled in $\ZFC$ and does not require $\b < {\aleph}_ω$, while an additional PCF type hypothesis, which holds when $\b < {\aleph}_ω$ 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{š}{á}k and Garc{\'ı}a Ferreira \cite{Hr1}, who applied it to the Katétov order on almost disjoint families.