Superstability in Tame Abstract Elementary Classes

In this paper we address a problem posed by Shelah in 1999 to find a suitable notion for superstability for abstract elementary classes in which limit models of cardinality $\mu$ are saturated. Theorem 1. Suppose that $\mathcal{K}$ is a $\chi$-tame abstract elementary class with no maximal models satisfying the joint embedding property and the amalgamation property. Suppose $\mu$ is a cardinal with $\mu \ge {\mathrm{\beth }}_{(2^{LS(\mathcal{K})+\chi }{)}^{+}}$. Let $M$ be a model of cardinality $\mu$. If $\mathcal{K}$ is both $\chi$-stable and $\mu$-stable and satisfies the $\mu$-superstability assumptions, then any two $\mu$-limit models over $M$ are isomorphic over $M$. Moreover, we identify sufficient conditions for superlimit models of cardinality $\mu$ to exist, for model homogeneous models to be superlimit, and for a union of saturated models to be saturated.