Exact saturation in simple and NIP theories

TL;DR

This paper characterizes when simple and NIP theories have models that are exactly saturated at singular cardinals, revealing that all simple theories do, and that NIP theories are exactly saturated iff they are not distal.

Contribution

It proves that all simple theories have exact saturation under certain set-theoretic assumptions and characterizes NIP theories with exact saturation as precisely those that are not distal.

Findings

All simple theories have exact saturation under certain assumptions.

NIP theories have exact saturation iff they are not distal.

Provides a new characterization of distality in model theory.

Abstract

A theory $T$ is said to have exact saturation at a singular cardinal $\kappa$ if it has a $\kappa$-saturated model which is not $\kappa^{+}$-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.