On weakly tight families

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