On Ramsey properties of classes with forbidden trees

Abstract

Abstract. Let F be a set of relational trees and let Forb h (F) be the class of all structures that admit no homomorphism from any tree in F; all this happens over a fixed finite relational signature σ. There is a natural way to expand Forb h (F) by unary relations to an amalgamation class. This expanded class, enhanced with a linear ordering, has the Ramsey property. Both forbidden trees and Ramsey properties have previously been linked to the complexity of constraint satisfaction problems.

Ramsey classes: examples and constructions

“…For a recent application of the partite method to prove the Ramsey property for classes of structures given by homomorphically forbidden trees, see [19].…”

“…19 Let C be the class of all finite {E, <}-structures where E denotes an equivalence relation and < denotes a linear order. It is easy to verify that C has the amalgamation property.…”