article published yes Domagoj Bradač author Patryk Morawski author Benny Sudakov author Yuval Wigderson author 2d0023a0-1567-11f0-833d-d5c1e476d4b5 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. American Mathematical Society2026 eng 0002-9939 1088-6826 2412.1759910.1090/proc/17442 1543927-942 yes Bradač, D., Morawski, P., Sudakov, B., &#38; Wigderson, Y. (2026). Ordered Ramsey numbers of graphs with 𝑚 edges. <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society. <a href="https://doi.org/10.1090/proc/17442">https://doi.org/10.1090/proc/17442</a> Bradač D, Morawski P, Sudakov B, Wigderson Y. Ordered Ramsey numbers of graphs with 𝑚 edges. <i>Proceedings of the American Mathematical Society</i>. 2026;154(3):927-942. doi:<a href="https://doi.org/10.1090/proc/17442">10.1090/proc/17442</a> D. Bradač, P. Morawski, B. Sudakov, Y. Wigderson, Proceedings of the American Mathematical Society 154 (2026) 927–942. Bradač, Domagoj, et al. “Ordered Ramsey Numbers of Graphs with 𝑚 Edges.” <i>Proceedings of the American Mathematical Society</i>, vol. 154, no. 3, American Mathematical Society, 2026, pp. 927–42, doi:<a href="https://doi.org/10.1090/proc/17442">10.1090/proc/17442</a>. D. Bradač, P. Morawski, B. Sudakov, and Y. Wigderson, “Ordered Ramsey numbers of graphs with 𝑚 edges,” <i>Proceedings of the American Mathematical Society</i>, vol. 154, no. 3. American Mathematical Society, pp. 927–942, 2026. Bradač, Domagoj, Patryk Morawski, Benny Sudakov, and Yuval Wigderson. “Ordered Ramsey Numbers of Graphs with 𝑚 Edges.” <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society, 2026. <a href="https://doi.org/10.1090/proc/17442">https://doi.org/10.1090/proc/17442</a>. Bradač D, Morawski P, Sudakov B, Wigderson Y. 2026. Ordered Ramsey numbers of graphs with 𝑚 edges. Proceedings of the American Mathematical Society. 154(3), 927–942. 221672026-06-29T10:54:32Z2026-07-14T08:34:43Z