On Constrained Generic Expansions and Structural Ramsey Theory

It is reasonably well-known in model theory that expansions of countable countablycategorical structures are closely associated with certain compactly metrizable spaces and, further, that a generic orbit in such a space – in the sense of Baire category – corresponds to an expansion with a particularly well-behaved model theory (relative to the base structure). Results of this kind can be found in [2], [6],[3] and several other publications. The first contribution of this article lies in demonstrating that for any expansion B (by finitely many new relations) of a countable countably-categorical structure A, there is canonical א0-categorical “maximally” generic expansion of A constrained by B – an expansion that is Baire-generic in the appropriate space and realizes no configurations that are not already realized in B. The motivating example, for us, of an expansion B of A is the expansion induced by a coloring of copies of a given finite substructure of A – this arises in the formulation and analysis of the Ramsey Property of a class K of finite structures from which A is generated (as, of course, the generic model of K). A first application of our analysis of constrained generic expansions is proving an equivalence between the Ramsey Property for K and a natural Generic Infinitary Ramsey Property for the generic modelA. The latter is formulated in terms of generic colorings and elementary self-embeddings of A; in [1], the author proved a equivalence between the Ramsey Property and a somewhat ad hoc two-part version of the Generic Infinitary Ramsey Property, so this article refines that characterization. Using our result on recovering generic expansions, we develop yet another characterization of the Ramsey Property, this time separating it into a Ramsey Property for 1-element structures (a Pigeonhole Principle) and a “1-simpliciality” property that provides a natural framework for induction arguments for the Ramsey Property in certain classes. To demonstrate the efficacy of this characterization, we provide new proofs of the Ramsey Property for the classes (i) of finite vertex-ordered graphs and (ii) finite trees (equipped with a certain kind of linear order).