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Abstract. We prove that from categoricity in λ + we can get categoricity in all cardinals ≥ λ + in a χ-tame abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided λ > LS(K) and λ ≥ χ.For the missing case when λ = LS(K), we prove that K is totally categorical provided that K is categorical in LS(K) and LS(K)+ .

Abstract. 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: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) ≤ (2 LS(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 > (2 Hanf (K) ) + , then K is stable in every κ with κ µ = κ. E.g. under GCH we have that K Galois-stable in µ implies that K is Galois-stable in µ +n for all n < ω.

Abstract. We prove:Main Theorem: Let K be an abstract elementary class satisfying the joint embedding and the amalgamation properties. Let µ be a cardinal above the the Löwenheim-Skolem number of the class. Suppose K satisfies the disjoint amalgamation property for limit models of cardinality µ. If K is µ-Galois-stable, has no µ-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, then any two (µ, σ ℓ )-limits over M for (ℓ ∈ {1, 2}) are isomorphic over M .This

Abstract. In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent:Corollary 0.1. Let K be a tame AEC with a monster model. Assume that K is stable in a proper class of cardinals. The following are equivalent:(1) For all high-enough λ, K has no long splitting chains.(2) For all high-enough λ, there exists a good λ-frame on a skeleton of K λ .(3) For all high-enough λ, K has a unique limit model of cardinality λ.(4) For all high-enough λ, K has a superlimit model of cardinality λ.(5) For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated. (6) There exists µ such that for all high-enough λ, K is (λ, µ)-solvable.This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.