Call a family of sets $\mathcal{F} \subseteq [\omega]^\omega$ *maximal* if there does not exist some $X \in [\omega]^\omega \setminus \mathcal{F}$ such that $X$ is almost disjoint with all elements of $\mathcal{F}$. Let $\mathcal{A} \subseteq \mathcal{F}$ be an almost disjoint family. Does there always exist some $\mathcal{A} \subseteq \mathcal{A}' \subseteq \mathcal{F}$ such that $\mathcal{A}'$ is a mad (maximal almost disjoint) family?


I am also interested in the weaker question: Does every maximal family contain a mad family?



Extending almost disjoint family in a maximal set
Every family $\mathcal{F}$ containing $\omega$ itself as a member is trivially maximal, since no infinite set is almost disjoint from $\omega$. But the family could otherwise consist of an almost disjoint non-maximal family $\mathcal{A}$. That is, $\mathcal{F}=\mathcal{A}\cup\{\omega\}$ is a counterexample.


