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 $$\langle \lambda _i: i\le \alpha <\aleph _1\rangle $$??i:i≤?<?1? is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP$$(<\lambda _0)$$(<?0), there is an $$L_{\omega _1,\omega }$$L?1,?-sentence $$\psi $$? whose models form a pure AEC and(1)The models of $$\psi $$? satisfy JEP$$(<\lambda _0)$$(<?0), while JEP fails for all larger cardinals and AP fails in all infinite cardinals.(2)There exist $$2^{\lambda _i^+}$$2?i+ non-isomorphic maximal models of $$\psi $$? in $$\lambda _i^+$$?i+, for all $$i\le \alpha $$i≤?, 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 $$\aleph _0$$?0 are at least $$\beth _{\omega _1}$$??1. We show that although $$AP(\kappa )$$AP(?) for each $$\kappa $$? implies the full amalgamation property, $$JEP(\kappa )$$JEP(?) 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.