# Gaps in cardinalities of MAD families

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

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

Suppose we force to add $$\kappa$$ mutually generic Cohen reals to a model of $$\mathsf{CH}$$, where $$\kappa$$ is some cardinal with uncountable cofinality. In the extension, there are MAD families of cardinality $$\aleph_1$$ and cardinality $$\kappa = \mathfrak{c}$$, but there are no MAD families of any intermediate cardinality.
They prove that the set of all cardinals $$\kappa$$ such that there is a MAD family of size $$\kappa$$ can be almost completely arbitrary. For example, given any $$A \subseteq \omega \setminus \{0\}$$, there is a forcing extension in which $$A = \{ n < \omega :\, \text{there is a MAD family of size } \aleph_n\}$$.