Ramsey goodness of paths

TL;DR

This paper proves that long paths are Ramsey-good for all graphs H when the path length exceeds four times the size of H, confirming a longstanding conjecture in graph Ramsey theory.

Contribution

It establishes that paths of length at least four times the size of H are H-good, confirming a conjecture by Allen, Brightwell, and Skokan.

Findings

Paths of length at least 4|H| are H-good.

Confirms a conjecture on Ramsey goodness of paths.

Advances understanding of Ramsey numbers for paths.

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 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 $\sigma$ 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 short note we prove that $n$-vertex path $P_n$ is $H$-good for all $n\geq 4|H|$. This proves in a strong form a conjecture of Allen, Brightwell, and Skokan.