2586809.pdf (210.32 kB)
Classification of δ-Invariant Amalgamation Classes
journal contributionposted on 2013-11-15, 00:00 authored by Roman D. Aref'ev, John T. Baldwin, Marco Mazzucco
Hrushovski's generalization of the Fraisse construction has provided a rich source of examples in model theory, model theoretic algebra and random graph theory. The construction assigns to a dimension function s and a class K of finite (finitely generated) models a countable 'generic' structure. We investigate here some of the simplest possible cases of this construction. The class K will be a class of finite graphs; the dimension, 6 (A), of a finite graph A will be the cardinality of A minus the number of edges of A. Finally and significantly we restrict to classes which are s-invariant. A class of finite graphs is s-invariant if membership of a graph in the class is determined (as specified below) by the dimension and cardinality of the graph, and dimension and cardinality of all its subgraphs. Note that a generic graph constructed as in Hrushovski's example of a new strongly minimal set does not arise from a s-invariant class. We show there are countably many s-invariant (strong) amalgamation classes of finite graphs which are closed under subgraph and describe the countable generic models for these classes. This analysis provides co-stable generic graphs with an array of saturation and model completeness properties which belies the similarity of their construction. In particular, we answer a question of Baizhanov (unpublished) and Baldwin  and show that this construction can yield an co-stable generic which is not saturated. Further, we exhibit some co-stable generic graphs that are not model complete. Most of the definitions and notation are carried over from  or from Baldwin and Shi . The existence of this variety of examples for graphs with the simplest choice of dimension function shows the diversity of even the class of theories with trivial forking.
Publisher StatementThe original version is available through Association for Symbolic Logic at DOI:10.2307/2586809
PublisherJournal of Symbolic Logic