Ordered Ramsey numbers of graphs with 𝑚 edges

Given a vertex-ordered graph G G , the ordered Ramsey number r > ( G ) r_>(G) is the minimum integer N N such that every 2 2 -coloring of the edges of the complete ordered graph K N K_N contains a monochromatic ordered copy of G G . Motivated by a similar question posed by Erdős 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 G with m m edges and no isolated vertices. We prove that r > ( G ) ≤ e 10 9 m ( log ⁡ log ⁡ m ) 3 / 2 r_>(G) \leq e^{10^9 \sqrt {m} (\log \log m)^{3/2}} for any such G G , which is tight up to the ( log ⁡ log ⁡ m ) 3 / 2 (\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 m edges.