  Consider an a.e.c. (abstract elementary class), that is, a class K of models
with a partial order refining inclusion (submodel) which satisfy the most basic
properties of an elementary class. Our test question is trying to show that the
function dot I (lambda, K), counting the number of models in K of cardinality
lambda up to isomorphism, is "nice", not chaotic, even without assuming it is
sometimes 1, i.e. categorical in some lambda's. We prove here that for some
closed unbounded class C of cardinals we have (a), (b) or (c) where (a) for
every lambda in C of cofinality aleph_0, dot I (lambda, K) greater than or
equal to lambda, (b) for every lambda in C of cofinality aleph_0 and M belongs
to K_lambda, for every cardinal kappa greater than or equal to lambda there is
N_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 large
enough (equivalently lambda greater than or equal to beth_{delta_*} where
delta_* = (2^{LST(mathfrak k)})^+).
  Recall that an important difference of non-elementary classes from the
elementary case is the possibility of having models in K, even of large
cardinality, which are maximal, or just failing clause (b).
