Toward Categoricity for Classes with no Maximal Models

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. ∗ The first author’s research was partially supported by ‘BSF’ (USA-Israel); Publication no 635 in the first author’s publication list. Done 12/96-1/97/4-97