Ordered Ramsey numbers of graphs with 𝑚 edges

Given a vertex-ordered graph G, the ordered Ramsey number r<(G) is the minimum integer N such that every 2-coloring of the edges of the complete ordered graph KN contains a monochromatic ordered copy of G. Motivated by a similar question posed by Erd˝os and Graham [On partition theorems for finite graphs, Infinite and finite sets (Colloq., Keszthely, 1973), North-Holland, Amsterdam-London, pp. 515–527] in the unordered setting, we study the problem of bounding the ordered Ramsey number of any ordered graph G with m edges and no isolated vertices. We prove that r<(G) ≤ e109√m(log log m)3/2 for any such G, which is tight up to the (log log m)3/2 factor in the exponent. As a corollary, we obtain the corresponding bound for the oriented Ramsey number of a directed graph with m edges.