## 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:

 300000 Other combinatorial set theory 3e+17 Cardinal characteristics of the continuum

### Keywords:

almost disjoint family; selective ideal; Fréchet space

### Citations:

Zbl 0369.02041; Zbl 0646.03048; Zbl 0854.04004
Full Text: