SUPERCOMPACT MEASURES AND THE GALVIN PROPERTY

Abstract We study saturation properties of σ $\sigma $ sigma -complete measures on P κ ( λ ) $P_\kappa (\lambda )$ upper P Subscript kappa Baseline left parenthesis lamda right parenthesis , where λ $\lambda $ lamda can be either regular or singular. In particular, we prove that in contrast to Galvin’s theorem, the Galvin property from [6] fails for normal, fine ultrafilters on P κ ( λ ) $P_\kappa (\lambda )$ upper P Subscript kappa Baseline left parenthesis lamda right parenthesis , answering a question of the first author and Goldberg from [10]. We then provide several applications of our results: to ultrafilters on successors under U A $UA$ upper U upper A , we generalize a result of Gitik regarding density of ground model sets in supercompact Prikry extensions, and to generating sets of P κ ( λ ) $P_\kappa (\lambda )$ upper P Subscript kappa Baseline left parenthesis lamda right parenthesis measures. In the second part of the article, we study variations of the Galvin property suitable for ultrafilters over P κ ( λ ) $P_\kappa (\lambda )$ upper P Subscript kappa Baseline left parenthesis lamda right parenthesis , and generalize a result of Foreman–Magidor–Zeman [17, Theorem 1.2] on determinacy of filter games to the two-cardinal setting, answering a question of the first author and Gitman from [9].