Exact saturation in simple and NIP theories

Abstract

A theory $T$ is said to have exact saturation at a singular cardinal $\hat{I}º$ if it has a $\hat{I}º$-saturated model which is not $\hat{I}{º}^{+}$-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.