We provide here the first steps toward a Classification Theory of Abstract Elementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some λ greater than its Löwenheim-Skolem number. We study the degree to which amalgamation may be recovered, the behaviour of non µsplitting types. Most importantly, the existence of saturated models in a strong enough sense is proved, as a first step toward a complete solution to the Loś Conjecture for these classes. Further results are in preparation.Annotated Content §0 Introduction.[We link the present work to previous articles in the same field, and provide a large-scale picture in which the results of this paper fit.] §1 How much amalgamation is left?[Following [Sh 88], we prove that although amalgamation is not assumed, amalgamation bases are 'dense' for our purposes. We also prove the existence of Universal Extensions over amalgamation bases.] §2 Types and Splitting [We provide here the right notions of type for our context, and study the behaviour of non µ-splitting.] §3 Building the right kind of limits [We define the classes + K µ,α and variants, in order to study in depth different concepts of limit models, and of saturated models. We prove the existence of a good notion of saturated models.]

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

Large cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained that way ('Unfoldable cardinals') behave as a 'boundary' between properties consistent with 'V=L' and existence of indiscernibles. We also provide an 'embedding characterisation' of the unfoldable cardinals and study their preservation and destruction by various different forcings.

We study versions of limit models adapted to the context of metric abstract elementary classes. Under categoricity and superstability‐like assumptions, we generalize some theorems from . We prove criteria for existence and uniqueness of limit models in the metric context.