### Keisler's order has infinitely many classes

by Malliaris and Shelah. [MiSh:1050]

Israel J Math, 2018

We prove, in ZFC, that there is an infinite strictly descending
chain of classes of theories in Keisler's order.
Thus Keisler's order is infinite and not a well order. Moreover,
this chain occurs within the simple unstable theories, considered
model-theoretically tame. Keisler's order is a large scale
classification program in model theory, introduced in the 1960s,
which compares the complexity of theories. Prior to this paper,
it was thought to have finitely many classes, linearly ordered. The
model-theoretic complexity we find is witnessed by a very natural
class of theories, the n-free k-hypergraphs studied by Hrushovski.
Notably, this complexity reflects the difficulty of amalgamation and
appears orthogonal to forking.

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