Ordered Ramsey numbers of graphs with $m$ edges

TL;DR

This paper establishes an upper bound on the ordered Ramsey number for graphs with m edges, showing it grows roughly exponentially with the square root of m, and confirms the bound's near-optimality.

Contribution

The paper provides a tight upper bound on the ordered Ramsey number for graphs with m edges, extending the understanding of Ramsey theory in ordered graph settings.

Findings

Bound on ordered Ramsey number: $r_<(G) \\leq e^{10^9 \\sqrt{m} (\\log \\log m)^{3/2}}$

Bound is tight up to the $(\log \log m)^{3/2}$ factor

Applies to both undirected and directed graphs with m edges

Abstract

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 $K_N$ contains a monochromatic ordered copy of $G$. Motivated by a similar question posed by Erd\H{o}s and Graham 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 $r_<(G) \leq e^{10^9 \sqrt{m} (\log \log m)^{3/2}}$ 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.