## 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.

### MSC:

 3e+15 Descriptive set theory 3e+35 Consistency and independence results 3e+17 Cardinal characteristics of the continuum

### Citations:

Zbl 0369.02041; Zbl 1325.03058; Zbl 1225.03059
Full Text: