Regular ultrafilters and finite square principles

Abstract

We show that many singular cardinals λ above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle □^fin_λ,D introduced in [3]. For such ultrafilters D and cardinals λ there are models of size λ for which M^λ/D is not λ⁺⁺-universal and elementarily equivalent models M and N of size λ for which M^λ/D and N^λ/D are non-isomorphic. The question of the existence of such ultrafilters and models was raised in [1].