Ramsey properties of edge-labelled graphs via completions

Abstract

Motivated by applications in structural Ramsey theory, we describe “metric-like” classes of edge-labelled graphs, study their completion problems and find Ramsey expansions. They turn out to be general enough to incorporate most of the known Ramsey results for edge-labelled graphs under a common framework and also solve a problem of Conant on generalised metric spaces. As a corollary of understanding completions, one obtains homomorphism dualities for these classes.