Galois-stability for Tame Abstract Elementary Classes

We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper we explore stability results in this context. We assume that \K is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include: (1) Galois-stability above the Hanf number implies that \kappa(K) is less than the Hanf number. Where \kappa(K) is the parallel of \kapppa(T) for f.o. T. (2) We use (1) to construct Morley sequences (for non-splitting) improving previous results of Shelah (from Sh394) and Grossberg & Lessmann. (3) We obtain a partial stability-spectrum theorem for classes categorical above the Hanf number.Comment: 23 page

Excellent Abstract Elementary Classes are tame

The assumption that an AEC is tame is a powerful assumption permitting development of stability theory for AECs with the amalgamation property. Lately several upward categoricity theorems were discovered where tameness replaces strong set-theoretic assumptions. We present in this article two sufficient conditions for tameness, both in form of strong amalgamation properties that occur in nature. One of them was used recently to prove that several Hrushovski classes are tame. This is done by introducing the property of weak $(\mu,n)$-uniqueness which makes sense for all AECs (unlike Shelah's original property) and derive it from the assumption that weak (\LS(\K),n)-uniqueness, (\LS(\K),n)-symmetry and (\LS(\K),n)-existence properties hold for all $n<\omega$. The conjunction of these three properties we call \emph{excellence}, unlike \cite{Sh 87b} we do not require the very strong (\LS(\K),n)-uniqueness, nor we assume that the members of \K are atomic models of a countable first order theory. We also work in a more general context than Shelah's good frames.Comment: 26 page