Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. We say $a,b\in [\omega]^\omega$ are *almost disjoint* if $a\cap b$ is finite. A subset $A\subseteq [\omega]^\omega$ is said to be an *almost disjoint family* if $a, b$ are almost disjoint for all $a \neq b \in A$. A standard application of Zorn's Lemma shows that every almost disjoint family is contained in a *maximal almost disjoint family (MAD family)* (maximal with respect to $\subseteq$).

A diagonalization argument shows that all infinite MAD families have uncountable cardinality. By ${\frak a}$ we denote the minimum cardinality that a MAD family can have. It is consistent that ${\frak a} < {\frak c} = 2^{\aleph_0}$.

**Question.** Is it consistent that

- ${\frak a} < {\frak c}$,
- there is a MAD family $A\subseteq [\omega]^\omega$ with $|A| = {\frak c}$, and
- there is a cardinal ${\frak g}$ with ${\frak a} \in {\frak g} \in {\frak c}$ such that there is no MAD family with cardinality ${\frak g}$?