On ordered Ramsey numbers of bounded-degree graphs

Abstract

An ordered graph is a pair $\mathcal{G}=(G,\prec )$ where G is a graph and ≺ is a total ordering of its vertices. The ordered Ramsey number $\bar{\mathrm{R}}(\mathcal{G})$ 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 $d\ge 3$, almost every d-regular graph G satisfies $\bar{\mathrm{R}}(\mathcal{G})\ge \frac{n^{3/2-1/d}}{4\log n\log \log n}$ for every ordering $\mathcal{G}$ of G. In particular, there are 3-regular graphs G on n vertices for which the numbers $\bar{\mathrm{R}}(\mathcal{G})$ 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 $\bar{\mathrm{R}}(\mathcal{G})$ is linear in n.

We also show that almost every ordered matching $\mathcal{M}$ with n vertices and with interval chromatic number two satisfies $\bar{\mathrm{R}}(\mathcal{M})\ge c{n}^2/{\log }^{2}n$ for some absolute constant c.