Superstability in abstract elementary classes

Rami Grossberg, Sebastien Vasey

Published 2015-07-15Version 1

We prove that several definitions of superstability in abstract elementary classes (AECs) are equivalent under the assumption that the class is tame, has amalgamation, joint embedding, and arbitrarily large models. This partially answers questions of Shelah. $\mathbf{Theorem}$ Let $K$ be a tame AEC with amalgamation, joint embedding, and arbitrarily large models. Assume $K$ is stable. Then the following are equivalent: 1) For all high-enough $\lambda$, there exists $\kappa \le \lambda$ such that there is a good $\lambda$-frame on the class of $\kappa$-saturated models in $K_\lambda$. 2) For all high-enough $\lambda$, $K$ has a unique limit model of cardinality $\lambda$. 3) For all high-enough $\lambda$, $K$ has a superlimit model of cardinality $\lambda$. 4) For all high-enough $\lambda$, the union of a chain of $\lambda$-saturated models is $\lambda$-saturated. 5) There exists $\theta$ such that for all high-enough $\lambda$, $K$ is $(\lambda, \theta)$-solvable.