The joint embedding property and maximal models

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 〈λi : i ≤ α < א1〉 is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP(< λ0), there is an Lω1,ω-sentence ψ whose models form a pure AEC and (1) The models of ψ satisfy JEP(< λ0), while JEP fails for all larger cardinals and AP fails in all infinite cardinals. (2) There exist 2 + i non-isomorphic maximal models of ψ in λ+i , for all i ≤ α, but no maximal models in any other cardinality; and (3) ψ 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 א0 are at least iω1 . We show that although AP (κ) for each κ implies the full amalgamation property, JEP (κ) for each κ 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. We investigate in this paper the spectra of joint embedding and of maximal models for an Abstract Elementary Class (AEC), in particular for AEC defined by universal Lω1,ωsentences under substructure. Our main result provides a collection of bipartite graphs whose combinatorics allows us to construct for any given countable strictly increasing sequence of characterizable cardinals (λi), a sentence of Lω1,ω whose models have joint embedding below λ0 and 2 λi + -many maximal models in each λ+i , but arbitrarily large models. Two examples of such sequences (λi) are: (1) an enumeration of an arbitrary countable subset of the iα, α < ω1, and (2) an enumeration of an arbitrary countable subset of the אn, n < ω. We give precise definitions and more details in Section 1. In Section 2, we describe our basic combinatorics and the main constructions are in Section 3. We now provide some background explaining several motivations for this study. In first order logic, work from the 1950’s deduces syntactic characterizations of such properties as joint embedding and amalgamation via the compactness theorem. The syntactic conditions immediately yield that if these properties hold in one cardinality they hold in all cardinalities. For AEC this situation is vastly different. In fact, one major stream studies what are sometimes called Jónsson classes that satisfy: amalgamation, joint embedding, and have arbitrarily large models. (See, for example, [She99, GV06, Bal09] and a series of paper such as [GV06].) Without this hypothesis the properties must be parameterized and the relationship between, e.g. the Joint Embedding Property (JEP) holding in various cardinals, becomes a topic for study. In [Gro02] Grossberg conjectures the existence of a Hanf number for the Amalgamation Property (AP): a cardinal μ(λ) such that if an AEC with Löwenheim number λ has the AP in some cardinal greater than μ(λ) then it has the amalgamation property in all larger cardinals. Boney [Bon] makes great progress on this problem by showing that if κ is strongly compact and an AEC K is categorical in λ for some λ ≥ κ, then K has JEP and AP above κ. Baldwin and Boney, [BB14], show that if Date: February 19, 2015. 2010 Mathematics Subject Classification. Primary 03C48 Secondary 03C75, 03C52, 03C30.