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Let $\operatorname{non}(\mathcal{M})$ be the least cardinality of a non-meagre subset of the reals. Let $\mathfrak{a}$ be the least cardinality of an infinite maximal almost disjoint family (i.e. $\mathcal{A} \subseteq [\omega]^\omega$ such that $A \cap B$ is finite for all $A \neq B$ in $\mathcal{A}$, and $\mathcal{A}$ is maximal under inclusion in the collection of almost disjoint families).

Is $\operatorname{non}(\mathcal{M}) < \mathfrak{a}$ known to be consistent?

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    $\begingroup$ I completely forgot about this, but this question was already asked here and in the comments I hint towards this paper which answers your question. $\endgroup$
    – Johannes Schürz
    Commented Nov 24 at 18:03
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    $\begingroup$ What is still open (to my knowledege) is the consistency of i < a which would also answer your question, since cof(M) ≤ i is provable in ZFC. $\endgroup$
    – Johannes Schürz
    Commented Nov 24 at 18:08
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    $\begingroup$ @JohannesSchürz I see. Thanks! $\endgroup$
    – Clement Yung
    Commented Nov 24 at 18:10

1 Answer 1

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As mentioned by Johannes in the comments, this has been asked here and the following paper provides an affirmative answer to my problem:

Fischer, Vera; Mejia, Diego Alejandro, Splitting, bounding, and almost disjointness can be quite different, Can. J. Math. 69, No. 3, 502-531 (2017). ZBL1372.03094.

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