### Almost disjoint families and ultrapowers

#### Abstract

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.

#### Keywords

Ultrafilters, Ultrapowers, Almost disjoint families, Banach spaces, epsilon-separated sets, Measurable cardinals

#### Full Text:

4. [PDF]DOI: https://doi.org/10.4115/jla.2021.13.4

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Journal of Logic and Analysis ISSN: 1759-9008