Ramsey numbers of ordered graphs

An ordered graph is a pair G=(G,≺)G=(G,≺) where GG is a graph and ≺ is a total ordering of its vertices. The ordered Ramsey number R̄(G) is the minimum number NN such that every ordered complete graph with NN vertices and with edges colored by two colors contains a monochromatic copy of GG. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings MnMn on nn vertices for which R̄ (Mn) is superpolynomial in nn. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number R̄ (G) is polynomial in the number of vertices of GG if the bandwidth of GG is constant or if GG is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.