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

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