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

by Baldwin and Shelah. [BlSh:393]

J Symbolic Logic, 1995

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.

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