Ramsey goodness of cycles

TL;DR

This paper proves that large cycles are Ramsey good with respect to graphs having high chromatic number and large smallest color class, confirming a conjecture for such graphs.

Contribution

It establishes that sufficiently large cycles are H-good when H has high chromatic number and large minimum color class, advancing understanding of Ramsey numbers for cycles.

Findings

Large cycles are H-good for graphs H with high chromatic number and large minimum color class.

The result confirms a conjecture of Allen, Brightwell, and Skokan for a broad class of graphs.

The proof applies when cycle length exceeds 10^60 times the size of H.

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 $R(G,H) \geq (|G|-1)(\chi(H)-1)+\sigma(H)$, where $\chi(H)$ is the chromatic number of $H$ and $\sigma(H)$ is the size of the smallest color class in a $\chi(H)$-coloring of $H$. A graph $G$ is called $H$-good if $R(G,H)= (|G|-1)(\chi(H)-1)+\sigma(H)$. The notion of Ramsey goodness was introduced by Burr and Erd\H{o}s in 1983 and has been extensively studied since then. In this paper we show that if $n\geq 10^{60}|H|$ and $\sigma(H)\geq \chi(H)^{22}$ then the $n$-vertex cycle $C_n$ is $H$-good. For graphs $H$ with high $\chi(H)$ and $\sigma(H)$, this proves in a strong form a conjecture of Allen, Brightwell, and Skokan.