Excellent Abstract Elementary Classes are tame

Rami Grossberg, Alexei S. Kolesnikov

Published 2005-09-14Version 1

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.