Seven characterizations of non-meager P-filters
Andrea Medini

The following is joint work with Kenneth Kunen and Lyubomyr Zdomskyy. All filters
are assumed to be on omega, and we identify a filter with a subspace of 2^omega through characteristic functions. While Shelah showed that it is consistent that there are no P-points, it is a long standing open problem whether it is possible to construct a non-meager P-filter in ZFC. We will give several topological/combinatorial conditions that, for a filter on omega, are equivalent to being a non-meager P-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager P-filter. (Recall that a space X is countable dense homogeneous if for every pair (D,E) of countable dense subsets of X there exists a homeomorphism h of X such that h[D]=E.) This answers a question of Hernandez-Gutierrez and Hrusak. Along the way, we also strengthen a result of Miller.

Furthermore, we will show that the statement "Every non-meager filter contains a non-meager P-subfilter" is independent of ZFC (more precisely, it is a consequence of u<g and its negation is a consequence of Diamond). It follows from results of Hrusak and Van Mill that, under u<g, the only possibilities for the number of types of countable dense subsets of a non-meager filter are 1 and c.