Ordered Ramsey numbers of powers of paths

Abstract

Given two vertex-ordered graphs $G$ and $H$, the ordered Ramsey number $R_<(G,H)$ is the smallest $N$ such that whenever the edges of a vertex-ordered complete graph $K_N$ are red/blue-coloured, then there is a red (ordered) copy of $G$ or a blue (ordered) copy of $H$. Let $P_n^t$ denote the $t$-th power of a monotone path on $n$ vertices. The ordered Ramsey numbers of powers of paths have been extensively studied. We prove that there exists an absolute constant $C$ such that $R_<(K_s,P_n^t)\leq R(K_s,K_t)^{C} \cdot n$ holds for all $s,t,n$, which is tight up to the value of $C$. As a corollary, we obtain that there is an absolute constant $C$ such that $R_<(K_n,P_n^t)\leq n^{Ct}$. These results resolve a problem and a conjecture of Gishboliner, Jin and Sudakov. Furthermore, we show that $R_<(P_n^t,P_n^t)\leq n^{4+o(1)}$ for any fixed $t$. This answers questions of Balko, Cibulka, Kr\'al and Kyn\v{c}l, and of Gishboliner, Jin and Sudakov.