Almost disjoint families and ultrapowers

We prove estimates for the cardinality of set-theoretic ultrapowers in terms of the cardinality of almost disjoint families. Such results are then applied to obtain estimates for the density of ultrapowers of Banach spaces. We focus on the change of the behavior of the corresponding ultrapower when certain ‘‘completeness thresholds’’ of the relevant ultrafilter are crossed. Finally, we also provide an alternative characterization of measurable cardinals.

Mad families, P+(I)-ideals and ideal convergence

Let I be an ideal on ?, the notion of I-AD family was introduced in [3]. Analogous to the well studied ideal I(A) generated by almost disjoint families, we introduce and investigate the ideal I(I-A). It turns out that some properties of I(I-A) depends on the structure of I. Denoting by a(I) the minimum of the cardinalities of infinite I-MAD families, several characterizations for a(I) ? ?1 will be presented. Motivated by the work in [23], we introduce the cardinality s?,?(I), and obtain a necessary condition for s?,?(I) = s(I). As an application, we show finally that if a(I) ? s(I), then BW property coincides with Helly property.

MADNESS IN VECTOR SPACES

We consider maximal almost disjoint families of block subspaces of countable vector spaces, focusing on questions of their size and definability. We prove that the minimum infinite cardinality of such a family cannot be decided in ZFC and that the “spectrum” of cardinalities of mad families of subspaces can be made arbitrarily large, in analogy to results for mad families on ω. We apply the author’s local Ramsey theory for vector spaces [32] to give partial results concerning their definability.