A family $\mathcal{A}$ of infinite subsets of $\omega$ is called almost disjoint if for any two distinct sets $a, b \in \mathcal{A}$, the intersection $a\cap b$ is finite.
An almost disjoint family $\mathcal{A}$ is called of true cardinality $\mathfrak{c}$ if for every $M \subseteq \omega$, the set $\{a\in \mathcal{A}:|a \cap M| = \aleph _0\}$ is either finite or it has size $\mathfrak{c}$ ($|\mathbb{R}| = \mathfrak{c}$).
An almost disjoint family $\mathcal{A}$ is called completely separable if for each $B\subseteq \omega$ such that the set $\{a\in \mathcal{A}: |a \cap B| = \aleph _0\}$ is infinite, there is some $a \in \mathcal{A}$ with $a \subseteq B$.
The definition of almost disjoint families of true cardinality $\mathfrak{c}$ appears in the article ``completely separable MAD families" by Michael Hrusak and Petr Simon (Link here) but apparently it's been known in Prague since the 70's or 80's.
Almost disjoint families of true cardinality $\mathfrak{c}$ are related with completely separable almost disjoint families (which were introduced by Hechler in 1971 in Classifying almost-disjoint families with applications to $\beta N - N$, Israel J. Math. 10 (1971), 413-432) as follows:
- Every completely separable almost disjoint family is of true cardinality $\mathfrak{c}$.
- Given an almost disjoint family is of true cardinality $\mathfrak{c}$, it is possible to build a completely separable almost disjoint family.
Question: Who should be attributed for the definition of almost disjoint families of true cardinality $\mathfrak{c}$?
