### 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 ofultrapowers. 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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