Ordered Ramsey numbers of graphs with 𝑚 edges Bradač, Domagoj Morawski, Patryk Sudakov, Benny Wigderson, Yuval 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 Society 2026 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/22167 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> eng info:eu-repo/semantics/altIdentifier/doi/10.1090/proc/17442 info:eu-repo/semantics/altIdentifier/issn/0002-9939 info:eu-repo/semantics/altIdentifier/e-issn/1088-6826 info:eu-repo/semantics/altIdentifier/arxiv/2412.17599 info:eu-repo/semantics/openAccess