The Joint Embedding Property and Maximal Models

TL;DR

This paper constructs a class of models demonstrating specific joint embedding and maximality properties within pure AECs, establishing lower bounds for Hanf numbers and illustrating limitations of combinatorial methods.

Contribution

It introduces pure AECs to analyze joint embedding and maximal models, providing new examples and bounds for Hanf numbers in model theory.

Findings

Existence of models with specific JEP and AP properties

Large numbers of non-isomorphic maximal models in certain cardinals

Hanf number for JEP and maximality is at least b_{\u03c9_1}

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 3.5.2 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.