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

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