The joint embedding property and maximal models

Abstract

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 $\langle \lambda _i: i\le \alpha <\aleph _1\rangle$ 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 Löwenheim 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 prove the main combinatorial device of this paper cannot be used to extend the main theorem to a complete sentence.