On cuts in ultraproducts of linear orders II

Abstract

We continue our study of the class $\mathscr{C}(D)$, where $D$ is a uniform ultrafilter on a cardinal $\kappa$ and $\mathscr{C}(D)$ is the class of all pairs $(\theta_1, \theta_2),$ where $(\theta_1, \theta_2)$ is the cofinality of a cut in $J^\kappa /D$ and $J$ is some $(\theta_1+\theta_2)^+$-saturated dense linear order. We give a combinatorial characterization of the class $\mathscr{C}(D)$. We also show that if $(\theta_1, \theta_2) \in \mathscr{C}(D)$ and $D$ is $\aleph_1$-complete or $\theta_1 + \theta_2 > 2^\kappa,$ then $\theta_1=\theta_2.$