# Sh:1030

• Malliaris, M., & Shelah, S. (2016). Existence of optimal ultrafilters and the fundamental complexity of simple theories. Adv. Math., 290, 614–681.
• Abstract:
We characterize the simple theories in terms of saturation of ultrapowers. This gives a dividing line at simplicity in Keisler’s order and gives a true outside definition of simple theories. Specifically, for any \lambda \geq \mu \geq \theta \geq \sigma such that \lambda = \mu^+, \mu = \mu^{<\theta} and \sigma is uncountable and compact (natural assumptions given our prior work, which allow us to work directly with models), we define a family of regular ultrafilters on \lambda called optimal, prove that such ultrafilters exist and prove that for any \mathcal{D} in this family and any M with countable signature, M^\lambda/\mathcal{D} is \lambda^+-saturated if Th(M) is simple and M^\lambda/ \mathcal{D} is not \lambda^+-saturated if Th(M) is not simple. The proof lays the groundwork for a stratification of simple theories according to the inherent complexity of coloring, and gives rise to a new division of the simple theories: (\lambda, \mu, \theta)-explicitly simple.
• Version 2015-11-25_11 (58p) published version (68p)
Bib entry
@article{Sh:1030,
author = {Malliaris, Maryanthe and Shelah, Saharon},
title = {{Existence of optimal ultrafilters and the fundamental complexity of simple theories}},
}