Hypergraphs of prime models and distributions of countable models of small theories

Hypergraphs of prime models over realizations of types in small theories are defined. On the basis of graph structures of models of small theories, hierarchies of sets in these hypergraphs, revealing structural connections in countable models of small theories, are established. The key role of graph-theoretic objects in constructions of Ehrenfeucht theories is proved. Using hypergraph constructions, a classification of complete first-order theories with finite Rudin-Keisler preorders is generalized to the class of all small theories. © 2010 Springer Science+Business Media, Inc.