### A dividing line within simple unstable theories

by Malliaris and Shelah. [MiSh:999]

Advances in Math, 2013

We give the first (ZFC) dividing line in Keisler's order among
the unstable theories, specifically among the simple unstable
theories.
That is, for any infinite cardinal lambda for which there is
mu <
lambda <= 2^mu, we construct a regular ultrafilter {D}
on lambda so that (i) for any model M of a stable theory or of
the random graph, M^lambda / {D} is lambda^+-saturated but
(ii) if Th(N) is not simple or not low then N^lambda / {d}
is not lambda^+-saturated. The non-saturation result relies on
the
notion of flexible ultrafilters. To prove the saturation result
we
develop a property of a class of simple theories, called Qr_1,
generalizing the fact that whenever B is a set of parameters
in some
sufficiently saturated model of the random graph, |B| = lambda
and
mu < lambda <= 2^mu, then there is a set A with |A| = mu
so that any nonalgebraic p in S(B) is finitely realized in A
.
In addition to giving information about simple unstable theories,
our proof reframes the problem of saturation of ultrapowers in
several
key ways. We give a new characterization of good filters in terms
of
``excellence,'' a measure of the accuracy of the quotient Boolean
algebra. We introduce and develop the notion of {moral} ultrafilters
on Boolean algebras. We prove a so-called ``separation of variables''
result which shows how the problem of constructing ultrafilters
to have
a precise degree of saturation may be profitably separated into
a more
set-theoretic stage, building an excellent filter, followed by
a more
model-theoretic stage: building so-called moral ultrafilters
on the
quotient Boolean algebra, a process which highlights the complexity
of
certain patterns, arising from first-order formulas, in certain
Boolean algebras.

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