  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 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.
