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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.
The finite models of a universal sentence $Phi$ are the age of a structure if and only if $Phi$ has the joint embedding property. We prove that the computational problem whether a given universal sentence $Phi$ has the joint embedding property is und
In [13] the authors show that if $mu$ is a strongly compact cardinal, $K$ is an Abstract Elementary Class (AEC) with $LS(K)<mu$, and $K$ satisfies joint embedding (amalgamation) cofinally below $mu$, then $K$ satisfies joint embedding (amalgamation)
Soare proved that the maximal sets form an orbit in $mathcal{E}$. We consider here $mathcal{D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer. Some orbits of $mathcal{D}$-maximal sets are well understood, e.g., hemim
Based on Hrushovski, Palac{i}n and Pillays example [6], we produce a new structure without the canonical base property, which is interpretable in Baudischs group. Said structure is, in particular, CM-trivial, and thus at the lowest possible level of the ample hierarchy.
An element $a$ of a lattice cups to an element $b > a$ if there is a $c < b$ such that $a cup c = b$. An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary degrees th