The (t−1) $(t-1)$‐chromatic Ramsey number for paths

Abstract

The following relaxation of the classical problem of determining the Ramsey number of a fixed graph has first been proposed by Erdős, Hajnal and Rado over 50 years ago. Given a graph G $G$ and an integer t ≥ 2 $t\ge 2$ determine the minimum number N $N$ such that in any t $t$‐coloured complete graph on N $N$ vertices there is a copy of G $G$ using only edges of some t − 1 $t-1$ colours. We determine the answer precisely when G $G$ is a path.