Abstract Classes with Few Models Have 'Homogeneous-Universal' Models

Classes with few models have 'homogeneous-universal' models 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' to serve as a universal domain for K. In the original constructions of Jónsson and Fraissè, K was a universal class and ordinary substructure played the role of ≤. Working with a cardinal λ satisfying λ <λ = λ 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 (chain homogeneous-universal) to replace the natural notion of (K, ≤)-homogeneous-universal model. Modulo a weak extension of ZFC (provable if V=L), we show (Corollary5.24) that a class K obeying certain minimal restrictions satisfies a fundamental dichotomy: For arbitrarily large λ, either K has the maximal number of models in power λ or K has a unique chain