Ordered Ramsey numbers

TL;DR

Ordered Ramsey numbers $r_{<}(H)$ extend classical Ramsey theory by requiring monochromatic copies to preserve vertex ordering. The paper demonstrates a striking separation from the unordered setting: even simple matchings can have superpolynomial $r_{<}$ for some orderings, while providing general upper bounds that depend on degeneracy and interval chromatic number. It establishes a broad framework linking ordered graphs to hypergraph Ramsey theory, yields off-diagonal results, and characterizes when ordered Ramsey numbers are linear in size via vertex covers, while proposing numerous open questions and directions for further study.

Abstract

Given a labeled graph $H$ with vertex set $\{1, 2,\ldots,n\}$, the ordered Ramsey number $r_<(H)$ is the minimum $N$ such that every two-coloring of the edges of the complete graph on $\{1, 2, \ldots,N\}$ contains a copy of $H$ with vertices appearing in the same order as in $H$. The ordered Ramsey number of a labeled graph $H$ is at least the Ramsey number $r(H)$ and the two coincide for complete graphs. However, we prove that even for matchings there are labelings where the ordered Ramsey number is superpolynomial in the number of vertices. Among other results, we also prove a general upper bound on ordered Ramsey numbers which implies that there exists a constant $c$ such that $r_<(H) \leq r(H)^{c \log^2 n}$ for any labeled graph $H$ on vertex set $\{1,2, \dots, n\}$.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Corollary 1.7
• Theorem 1.8
• Theorem 2.1
• proof