Existence of optimal ultrafilters and the fundamental complexity of simple theories

by Malliaris and Shelah. [MiSh:1030]
Advances in Math, 2016
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 >= mu >= theta >= 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 emph {optimal}, prove that such ultrafilters exist and prove that for any {D} in this family and any M with countable signature, M^lambda / {D} is lambda^+-saturated if Th(M) is simple and M^lambda / {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.

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