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Selectivity of almost disjoint families. (English) Zbl 0992.03053

In this interesting paper, the author studies Ramsey-theoretic properties of filters on \(\omega\). Specifically, a filter \(\mathcal F\) on \(\omega\) is called \(+\)-Ramsey if every \(\mathcal F^+\)-branching subtree of \(\omega^{<\omega}\) has a branch \(b\in \omega^\omega\) such that \(\{b(n) : n\in \omega\}\) is a member of \(\mathcal F^+\) where, as usual, \(\mathcal F^+ \) is the set of all sets that meet each member of \(\mathcal F\). The notion of \(+\)-Ramsey is a strengthening of the more familiar notion of selective or happy families (see Zbl 0369.02041, Zbl 0646.03048 and Zbl 0854.04004). Of particular interest to the author is the case when \(\mathcal F\) is the dual filter to the ideal generated by some MAD (maximal almost disjoint) family \(\mathcal A\); in such a case, the family \(\mathcal A\) is said to be \(+\)-Ramsey. There is a nice construction to show that not every MAD family is \(+\)-Ramsey and it is shown that \(+\)-Ramsey MAD families exist if, for example, the covering of the meager ideal is \(\mathfrak c\). The author poses the question of whether there is a \(+\)-Ramsey MAD family in ZFC. In the final section, the author connects the notion of \(+\)-Ramsey with Arkhangelskij’s \(\alpha_i\)-properties of Fréchet spaces. If \(\mathcal F\) is the neighborhood filter of a point \(x\) in a countable Fréchet space \(\{x\}\cup \omega\), then if \(x\) is an \(\alpha_2\)-point, \(\mathcal F\) is \(+\)-Ramsey, and if \(\mathcal F\) is \(+\)-Ramsey, \(x\) is an \(\alpha_4\)-point. It is also shown that the existence of a \(+\)-Ramsey MAD family gives rise to a \(+\)-Ramsey filter \(\mathcal F\) on \(\{x\}\cup \omega\) which is Fréchet \(\alpha_4\) but not \(\alpha_3\).

MSC:

03E05 Other combinatorial set theory
03E17 Cardinal characteristics of the continuum
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