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 KN 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) e^10⁹ {m (m) ^3/2} for any such G G, which is tight up to the (log log m) 3 / 2 (m) ^3/2
Bradač et al. (Fri,) studied this question.