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](https://en.wikipedia.org/wiki/Zorn%27s_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](https://www.math.uni-hamburg.de/home/geschke/papers/IndependentFamilies.pdf) 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 ${\frak g}$?



Gaps in cardinalities of MAD families
Yes, this is consistent.


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.


The proof of this is a basic instance of what's known as an "isomorphism of names" argument. I think this result appears as an exercise in Kunen's 1980 book, although I think it's originally due to Arnie Miller. The argument is greatly extended in



> 
> S. Shelah and O. Spinas, "MAD spectra," *Journal of Symbolic Logic* **80** (2015), pp. 243-262 ([link](https://arxiv.org/abs/1402.5616)).
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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 $A = \{ n < \omega :\, \text{there is a MAD family of size } \aleph\_n\}$.


