Ramsey goodness of cycles

Abstract

Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it is well known and easy to show that $R(G,H)\ge (|G|-1)(\ddot{I}‡(H)-1)+\ddot{I}ƒ(H)$, where $\ddot{I}‡(H)$ is the chromatic number of $H$ and $\ddot{I}ƒ(H)$ is the size of the smallest color class in a $\ddot{I}‡(H)$-coloring of $H$. A graph $G$ is called $H$-good if $R(G,H)=(|G|-1)(\ddot{I}‡(H)-1)+\ddot{I}ƒ(H)$. The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively studied since then. In this paper we show that if $n\ge 10^{60}|H|$ and $\ddot{I}ƒ(H)\ge \ddot{I}‡(H{)}^{22}$ then the $n$-vertex cycle $C_n$ is $H$-good. For graphs $H$ with high $\ddot{I}‡(H)$ and $\ddot{I}ƒ(H)$, this proves in a strong form a conjecture of Allen, Brightwell, and Skokan.