Upward Stability Transfer for Tame Abstract Elementary Classes

Grossberg and VanDieren have started a program to develop a stability theory for tame classes (see [GrVa1]). We name some variants of tameness (Definitions 1.4 and 1.7) and prove the following. Theorem 0.1. Let K be an AEC with Löwenheim-Skolem number ≤ κ. Assume that K satisfies the amalgamation property and is κ-weakly tame and Galois-stable in κ. Then, K is Galois-stable in κ for all n < ω. With one further hypothesis we get a very strong conclusion in the countable case. Theorem 0.2. Let K be an AEC satisfying the amalgamation property and with Löwenheim-Skolem number א0 that is ω-local and א0-tame. If K is א0-Galois-stable then K is Galois-stable in all cardinalities.