On ordered Ramsey numbers of bounded-degree graphs

TL;DR

This paper investigates ordered Ramsey numbers for bounded-degree graphs, showing that for most 3-regular graphs the numbers grow superlinearly, while for degree-2 graphs they can be linear, and also explores bounds for ordered matchings.

Contribution

It establishes superlinear lower bounds for ordered Ramsey numbers of most 3-regular graphs and linear bounds for degree-2 graphs, solving a problem posed by Conlon, Fox, Lee, and Sudakov.

Findings

Almost all 3-regular graphs have superlinear ordered Ramsey numbers.

Degree-2 graphs can have linear ordered Ramsey numbers with appropriate ordering.

Ordered matchings with interval chromatic number two have at least quadratic ordered Ramsey numbers.

Abstract

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the edges of the ordered complete graph on $N$ vertices contains a monochromatic copy of $\mathcal{G}$. We show that for every integer $d \geq 3$, almost every $d$-regular graph $G$ satisfies $\overline{R}(\mathcal{G}) \geq \frac{n^{3/2-1/d}}{4\log{n}\log{\log{n}}}$ for every ordering $\mathcal{G}$ of $G$. In particular, there are 3-regular graphs $G$ on $n$ vertices for which the numbers $\overline{R}(\mathcal{G})$ are superlinear in $n$, regardless of the ordering $\mathcal{G}$ of $G$. This solves a problem of Conlon, Fox, Lee, and Sudakov. On the other hand, we prove that every graph $G$ on $n$ vertices with maximum degree 2 admits an ordering $\mathcal{G}$ of $G$ such that $\overline{R}(\mathcal{G})$ is linear in $n$. We also show that almost every ordered matching $\mathcal{M}$ with $n$ vertices and with interval chromatic number two satisfies $\overline{R}(\mathcal{M}) \geq cn^2/\log^2{n}$ for some absolute constant $c$.