  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.
