A Dividing Line Within Simple Unstable Theories

Abstract

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 $\mu < \lambda \leq 2^\mu$, we construct a regular ultrafilter D on $\lambda$ such 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 Qr1, generalizing the fact that whenever $B$ is a set of parameters in some sufficiently saturated model of the random graph, $|B| = \lambda$ and $\mu < \lambda \leq 2^\mu$, then there is a set $A$ with $|A| = \mu$ such that any non-algebraic $p \in S(B)$ 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 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.