A General Upper Bound on Multicolor Ordered Ramsey Numbers

Abstract

We provide a general upper bound on multicolor ordered Ramsey numbers in terms of the interval chromatic number and the degeneracy of an ordered graph. We extend previous results by Conlon, Fox, Lee, and Sudakov (2017) by showing that for every $n$-vertex ordered graph $G^<$ with degeneracy $d\geq2$, and interval chromatic number $χ$, its $q$-color ordered Ramsey number satisfies $r_<(G^<;q) \in n^{O(d^{q-1}\lceil \logχ\rceil^{q-1})}$ for every $q \geq 2$. For fixed parameters $q,d,χ$, the resulting estimate is polynomial in $n$. For triangle-free ordered graphs $G^<$, we also provide the stronger estimate $n^{O(q^2d {\lceil \log χ\rceil}^{q-1})}$. It also follows from a recent result by Li (2026) that our upper bound is almost tight for ordered matchings.