Ramsey goodness of 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 $R(G,H)\ge \left(\right|G|-1)\left(\chi \right(H)-1)+\sigma \left(H\right)$, where $\chi \left(H\right)$ is the chromatic number of H and σ the size of the smallest color class in a $\chi \left(H\right)$-coloring of H. A graph G is called H-good if $R(G,H)=\left(\right|G|-1)\left(\chi \right(H)-1)+\sigma \left(H\right)$. The notion of Ramsey goodness was introduced by Burr and Erdős 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 $n\ge 4\left|H\right|$. This proves in a strong form a conjecture of Allen, Brightwell, and Skokan.