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 ${\mathrm{\aleph }}_0$ are at least ${\mathrm{\beth }}_{{\omega }_1}$. 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.