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 3.5.2 and Corollary 3.5.6): If $(\lambda_i : i \le \alpha<\aleph_1)$ is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP$(<\lambda_0)$, there is an $L_{\omega_1,\omega}$ -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 $i \le \alpha$, 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 $\aleph_0$ are at least $\beth_{\omega_1}$. We show that although AP$(\kappa)$ for each $\kappa$ implies the full amalgamation property, JEP$(\kappa)$ for each \kappa does not imply the full joint embedding property. We show the main combinatorial device of this paper cannot be used to extend the main theorem to a complete sentence.