### Toward Categoricity for Classes with no Maximal Models

by Shelah and Villaveces. [ShVi:635]

Annals Pure and Applied Logic, 1999

We provide here the first steps toward Classification Theory of
Abstract Elementary Classes with no maximal models, plus some mild
set theoretical assumptions, when the class is categorical in some
lambda greater than its L{o}wenheim-Skolem number. We study the
degree to which amalgamation may be recovered, the behaviour of non
mu-splitting 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 Los Conjecture for these classes.

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