  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.
