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 new context. We assume that K is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include: Theorem 0.1. Suppose that K is not only tame, but LS(K)-tame. If μ ≥ Hanf(K) and K is Galois stable in μ, then κμ(K) < i(2Hanf(K))+ , where κμ(K) is a relative of κ(T ) from first order logic. Hanf(K) is the Hanf number of the class K. It is known that Hanf(K) ≤ i(2LS(K))+ The theorem generalizes a result from [Sh3]. It is used to prove both the existence of Morley sequences for non-splitting (improving Claim 4.15 of [Sh 394] and a result from [GrLe1]) and the following initial step towards a stability spectrum theorem for tame classes: Theorem 0.2. If K is Galois-stable in some μ > i(2Hanf(K))+ , then K is stable in every κ with κ = κ. E.g. under GCH we have that K Galois-stable in μ implies that K is Galois-stable in μ for all n < ω.