On ordered Ramsey numbers of bounded-degree graphs

An ordered graph is a pair G = (G,≺) where G is a graph and ≺ is a total ordering of its vertices. The ordered Ramsey number R(G) is the minimum number N such that every ordered complete graph with N vertices and with edges colored by two colors contains a monochromatic copy of G. We show that for every integer d ≥ 3, almost every d-regular graph G satisfies R(G) ≥ n3/2−1/d 4 log n log log n for every ordering G of G. In particular, there are 3-regular graphs G on n vertices for which the numbers R(G) are superlinear in n, regardless of the ordering G of G. This solves a problem of Conlon, Fox, Lee, and Sudakov. On the other hand, we prove that every graph G on n vertices with maximum degree 2 admits an ordering G of G such that R(G) is linear in n. We also show that almost every ordered matchingM with n vertices and with interval chromatic number two satisfies R(M) ≥ cn2/ log2 n for some absolute constant c.