Abstract classes with few models have `homogeneous-universal' models

Abstract

This paper is concerned with a class K of models and an abstract notion of submodel <=. Experience in first order model theory has shown the desirability of finding a `monster model&#39; to serve as a universal domain for K. In the original constructions of Jonsson and Fraisse, K was a universal class and ordinary substructure played the role of <=. Working with a cardinal lambda satisfying lambda^{< lambda}= lambda guarantees appropriate downward Lowenheim-Skolem theorems; the existence and uniqueness of a homogeneous-universal model appears to depend centrally on the amalgamation property. We make this apparent dependence more precise in this paper. The major innovation of this paper is the introduction of weaker notion to replace the natural notion of (K, <=)-homogeneous-universal model. Modulo a weak extension of ZFC (provable if V=L), we show that a class K obeying certain minimal restrictions satisfies a fundamental dichotomy: For arbitrarily large lambda, either K has the maximal number of models in power lambda or K has a unique chain homogenous-universal model of power lambda. We show that in a class with amalgamation this dichotomy holds for the notion of K-homogeneous-universal model in the more normal sense.