The (t-1)-chromatic Ramsey number for paths

TL;DR

This paper precisely determines the (t-1)-chromatic Ramsey number for paths, a relaxed version of classical Ramsey problems, extending understanding of colourings in complete graphs.

Contribution

The paper provides an exact solution for the (t-1)-chromatic Ramsey number specifically for paths, filling a gap in the understanding of colourings in Ramsey theory.

Findings

Exact (t-1)-chromatic Ramsey number for paths established

Extends classical Ramsey theory to (t-1)-colour constraints

Provides a basis for further research on other graph classes

Abstract

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