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A family $\mathcal A$ of infinite subsets of $\omega$ is called almost disjoint if for any distinct sets $A,B\in\mathcal A$ the intersection $A\cap B$ is finite.

Let $\mathfrak a'$ be the largest cardinal such that for any cardinal $\kappa<\mathfrak a'$ and any family of infinite sets $\{X_\alpha\}_{\alpha\in\kappa}\subseteq[\omega]^\omega$ there exists an almost disjoint family of infinite sets $(A_\alpha)_{\alpha\in\kappa}$ such that $A_\alpha\subseteq X_\alpha$ for all $\alpha\in\kappa$.

By Proposition 6.2 in this preprint, $$\mathfrak a\le\min\{\mathfrak a^+,\mathfrak c\}\le\mathfrak a'\le\mathfrak c,$$ where $\mathfrak a$ is the smallest cardinality of a maximal infinite almost disjoint family $\mathcal A\subseteq[\omega]^\omega$.

Problem. What can be said about the cardinal $\mathfrak a'$? Is it equal to $\mathfrak c$? Or maybe to $\min\{\mathfrak a^+,\mathfrak c\}$?

For problems motivating this question it would be desirable to have $\mathfrak a'>\mathfrak r$. So, let us ask

Question. Is $\mathfrak a'\le\mathfrak r<\mathfrak c$ consistent?

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  • $\begingroup$ Why is it obvious that $\mathfrak{a} \leq \mathfrak{a}'$? $\endgroup$
    – Johannes Schürz
    Commented Apr 16, 2020 at 23:28
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    $\begingroup$ @JohannesSchürz I agree, it is not so obvious. The proof of the inequality $\mathfrak a\le\mathfrak a′$ (even in the stronger form $\min\{\mathfrak a^+,\mathfrak c\}\le\mathfrak a′$ ) can be found in Proposition 6.2 of this preprint: researchgate.net/publication/… Now I will correct the question appropriately. Thank you for the comment. $\endgroup$
    – Taras Banakh
    Commented Apr 17, 2020 at 5:34
  • $\begingroup$ I'm sorry, you are right, it is pretty obvious. For whatever reason, I was missing that $\mathfrak{a}$ of $[\omega]^\omega$ is the same as $\mathfrak{a}_X$ of $[X]^\omega$ for every $X \in [\omega]^\omega$... $\endgroup$
    – Johannes Schürz
    Commented Apr 17, 2020 at 11:13
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    $\begingroup$ Yes, $\mathfrak{a}'=\mathfrak{c}$. This is a special case of Theorem 2.1 in the paper "Weak saturation properties of ideals", by J. Baumgartner, A. Hajnal and A. Maté, researchgate.net/publication/… $\endgroup$
    – Santi Spadaro
    Commented Apr 17, 2020 at 17:57
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    $\begingroup$ @SantiSpadaro Thank you, Santi, very much for the comment and the link. If you want, you can write it as an answer and I will accept it, which will allow to close this my question as answered. $\endgroup$
    – Taras Banakh
    Commented Apr 17, 2020 at 18:28

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