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 2-coloring of the edges of the ordered complete graph on N vertices contains a monochromatic copy of G. We show that for every integer d≥3, almost every d-regular graph G satisfies R‾(G)≥[Formula presented] 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 matching M with n vertices and with interval chromatic number two satisfies R‾(M)≥cn²/log²⁡n for some absolute constant c.