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 $\chi$, its $q$-color ordered Ramsey number satisfies $r_<(G^<;q) \in n^{O(d^{q-1}\lceil \log{\chi}\rceil^{q-1})}$ for every $q \geq 2$. For fixed parameters $q,d,\chi$, the resulting estimate is polynomial in $n$. For triangle-free ordered graphs $G^<$, we also provide the stronger estimate $n^{O(q^2d {\lceil \log \chi \rceil}^{q-1})}$. It also follows from a recent result by Li (2026) that our upper bound is almost tight for ordered matchings.
The \emph{ordered Ramsey number} $r_<(G,H)$ of ordered $k$-graphs $G$ and $H$ is the least integer $N$ such that every red-blue edge-coloring of the naturally ordered complete $k$-graph on $[N]$ contains a blue ordered copy of $G$ or a red ordered copy of $H$. We prove that there is an absolute constant $c>0$ such that...
Wen Chen, Zi-Han He, Qi-Zhong Lin et al.· 0 citations
For an ordered graph $H$ and an integer $q \geq 2$, let $r_{<}(H;q)$ denote the $q$-color ordered Ramsey number of $H$. Conlon, Fox, Lee and Sudakov asked whether, for every $q \geq 3$, there is a constant $c_q$ such that $r_{<}(M;q) \leq n^{c_q\log n}$ for every ordered matching $M$ on $n$ vertices. We answer this neg...
We prove the following two upper bounds for ordered Ramsey numbers: (1) Every ordered forest $F$ on $n$ vertices satisfies $R_{<}(F,F)=O(n^{1+\lceil\log\chi_{<}(F)\rceil})$. This in particular answers a question of Geneson, Holmes, Liu, Neidinger, Pehova and Wass. (2) There is a function $f$ such that, for every fixed...
We prove that there is an absolute positive constant $c$ such that every bipartite graph with $N$ vertices in each part and at least $N^2/2$ edges contains a complete bipartite graph $K_{t,t}$ whenever $N\ge c\, 2^{t}$. This improves the classical K\H{o}v\'ari-S\'os-Tur\'an bound requiring $N$ of order $t\,2^t $. As a...
For ordered graphs $H_1,\ldots,H_t$, let $\rt(H_1,\ldots,H_t)$ denote the least integer $N$ such that every $t$-coloring of the edges of the naturally ordered complete graph on $[N]$ contains an ordered copy of $H_i$ in color $i$ for some $i\in[t]$. We prove that a uniformly random ordered matching $M$ on $n$ vertices...
The celebrated result of Johansson, Kahn and Vu determined the threshold order for clique factors in random graphs, and subsequent work identified the sharp threshold and the corresponding hitting-time phenomenon. In this paper we study the probability that there is no $K_r$-factor above the threshold and, more general...
Zhi-Fei Yan· 0 citations
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