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**Mad families constructed from perfect almost disjoint families.**
*(English)*
Zbl 1375.03057

An infinite family \(A\subseteq [\omega]^\omega\) is called maximal almost disjoint (or mad) if any two elements of \(A\) have finite intersection and \(\forall b\in[\omega]^\omega\,\exists a\in A\,|a\cap b|=\omega\). It was proved in [A. R. D. Mathias, Ann. Math. Logic 12, 59–111 (1977; Zbl 0369.02041)] that there are no analytic mad families. Later on, A. Törnquist [J. Symb. Log. 78, No. 4, 1181–1182 (2013; Zbl 1325.03058)] showed that the existence of a coanalytic mad family is equivalent to the existence of a \(\boldsymbol{\Sigma}^1_2\) mad family. Using forcing techniques, it was proved that the existence of a \(\boldsymbol{\Sigma}^1_2\) mad family is consistent with certain set-theoretical assumptions.

In the paper under review, the authors investigate mad families in models obtained by adding dominating reals. In [S.-D. Friedman and L. Zdomskyy, Ann. Pure Appl. Logic 161, No. 12, 1581–1587 (2010; Zbl 1225.03059)], it was shown that the existence of a \(\boldsymbol{\Pi}^1_2\) \(\omega\)-mad family is consistent with \(\mathfrak{b}>\aleph_1\) and asked whether a better result is possible for the more general case of a mad family. This question is answered positively in the reviewed paper. Namely, it is proved that the existence of a \(\boldsymbol{\Pi}^1_1\) mad family is consistent with \(\mathfrak{b}>\aleph_1\). To obtain this result, the authors introduce a new approach to preservation: rather than constructing a mad family \(A\) whose maximality is preserved directly, they construct \(A\) as a union of \(\aleph_1\)-many Borel sets in such a way that the union of the same sets re-interpreted in the larger model remains a mad family. They also isolate a new cardinal invariant – the Borel almost-disjointness number \(\mathfrak{a}_B\) and proves that \(\mathfrak{a}_B<\mathfrak{b}\) is consistent.

In the paper under review, the authors investigate mad families in models obtained by adding dominating reals. In [S.-D. Friedman and L. Zdomskyy, Ann. Pure Appl. Logic 161, No. 12, 1581–1587 (2010; Zbl 1225.03059)], it was shown that the existence of a \(\boldsymbol{\Pi}^1_2\) \(\omega\)-mad family is consistent with \(\mathfrak{b}>\aleph_1\) and asked whether a better result is possible for the more general case of a mad family. This question is answered positively in the reviewed paper. Namely, it is proved that the existence of a \(\boldsymbol{\Pi}^1_1\) mad family is consistent with \(\mathfrak{b}>\aleph_1\). To obtain this result, the authors introduce a new approach to preservation: rather than constructing a mad family \(A\) whose maximality is preserved directly, they construct \(A\) as a union of \(\aleph_1\)-many Borel sets in such a way that the union of the same sets re-interpreted in the larger model remains a mad family. They also isolate a new cardinal invariant – the Borel almost-disjointness number \(\mathfrak{a}_B\) and proves that \(\mathfrak{a}_B<\mathfrak{b}\) is consistent.

Reviewer: Jan Kraszewski (Wrocław)