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Consider an a.e.c. (abstract elementary class), that is, a class K of modelswith a partial order refining inclusion (submodel) which satisfy the most basicproperties of an elementary class. Our test question is trying to show that thefunction dot I (lambda, K), counting the number of models in K of cardinalitylambda up to isomorphism, is "nice", not chaotic, even without assuming it issometimes 1, i.e. categorical in some lambda's. We prove here that for someclosed unbounded class C of cardinals we have (a), (b) or (c) where (a) forevery lambda in C of cofinality aleph_0, dot I (lambda, K) greater than orequal to lambda, (b) for every lambda in C of cofinality aleph_0 and M belongsto K_lambda, for every cardinal kappa greater than or equal to lambda there isN_kappa of cardinality kappa extending M (in the sense of our a.e.c.), (c)mathfrak k is bounded; that is, dot I (lambda, K) = 0 for every lambda largeenough (equivalently lambda greater than or equal to beth_{delta_*} wheredelta_* = (2^{LST(mathfrak k)})^+).  Recall that an important difference of non-elementary classes from theelementary case is the possibility of having models in K, even of largecardinality, which are maximal, or just failing clause (b).

A.E.C. with not too many models