The joint embedding property and maximal models

We introduce the notion of a 'pure' Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.34 and Corollary 3.38): If <lambda(i) : i <= alpha < N-1 > is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whosemodels satisfy JEP(< lambda(0)), there is an L-omega 1,L-omega-sentence psi whose models form a pure AEC and (1) The models of psi satisfy JEP (< lambda(0)), while JEP fails for all larger cardinals and AP fails in all infinite cardinals. (2) There exist 2(lambda i+) non-isomorphic maximal models of psi in lambda(+)(i), for all i <= alpha, but no maximal models in any other cardinality; and (3) psi has arbitrarily large models. In particular this shows the Hanf number for JEP and the Hanf number for maximality for pure AEC with Lowenheim number N-0 are at least beth(omega 1). We show that although AP(kappa) for each kappa implies the full amalgamation property, J EP(kappa) for each kappa does not imply the full joint embedding property. We prove the main combinatorial device of this paper cannot be used to extend the main theorem to a complete sentence.