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 $λ$ for which there is $μ< λ\leq 2^μ$, we construct a regular ultrafilter D on $λ$ such that (i) for any model $M$ of a stable theory or of the random graph, $M^λ/D$ is $λ^+$-saturated but (ii) if $Th(N)$ is not simple or not low then $N^λ/D$ is not $λ^+$-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, $|B| = λ$ and $μ< λ\leq 2^μ$, then there is a set $A$ with $|A| = μ$ 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.
$n$-variable theorems for dividing lines characterized by positive consistency-inconsistency configurations, and their applications to preservation problems
