On ordered Ramsey numbers of bounded-degree graphs

An ordered graph is a pair G = (G, (sic)) where G is a graph and is a total ordering of its vertices. The ordered Ramsey number (R) over bar (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) over bar (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) over bar (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) over bar (M) >= cn(2)/ log(2) n for some absolute constant c. (C) 2018 Elsevier Inc. All rights reserved.