Keisler's order is not simple (and simple theories may not be either)

TL;DR

This paper demonstrates that Keisler's order on theories has the maximum number of classes, revealing complex structure even among simple unstable theories with no nontrivial forking.

Contribution

It shows that Keisler's order is not simple and constructs theories with maximal complexity, combining model theory, set theory, and combinatorics.

Findings

Keisler's order has the maximum number of classes.

Constructed simple unstable theories with no nontrivial forking.

Connected growth rates of sequences to regular pairs in Szemerédi's lemma.

Abstract

Solving a decades-old problem we show that Keisler's 1967 order on theories has the maximum number of classes. The theories we build are simple unstable with no nontrivial forking, and reflect growth rates of sequences which may be thought of as densities of certain regular pairs, in the sense of Szemer\'edi's regularity lemma. The proof involves ideas from model theory, set theory, and finite combinatorics.