On Ramsey properties of classes with forbidden trees

Let F be a set of relational trees and let Forbh(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 $\sigma$. There is a natural way to expand Forbh(F) by unary relations to an amalgamation class. This expanded class, enhanced with a linear ordering, has the Ramsey property.Comment: Keywords: forbidden substructure; amalgamation; Ramsey class; partite method v2: changed definition of expanded class; v3: final versio

Decidable Relationships between Consistency Notions for Constraint Satisfaction Problems

Abstract. We define an abstract pebble game that provides game interpretations for essentially all known consistency algorithms for constraint satisfaction problems including arc-consistency, (j, k)-consistency, k-consistency, k-minimality, and refinements of arc-consistency such as peek arc-consistency and singleton arc-consistency. Our main result is that for any two instances of the abstract pebble game where the first satisfies the additional condition of being stacked, there exists an algorithm to decide whether consistency with respect to the first implies consistency with respect to the second. In particular, there is a decidable criterion to tell whether singleton arc-consistency with respect to a given constraint language implies k-consistency with respect to the same constraint language, for any fixed k. We also offer a new decidable criterion to tell whether arc-consistency implies satisfiability which pulls from methods in Ramsey theory and looks more amenable to generalization.