Multicoloured Ramsey numbers of the path of length four

TL;DR

This paper determines the multicoloured Ramsey number for the path of length four, showing it is approximately three times the number of colours, advancing understanding of colourings in complete graphs.

Contribution

The paper explicitly calculates the multicoloured Ramsey number for the path of length four, filling a gap in combinatorial graph theory.

Findings

$R_r(P_5)$ is approximately $3r$

Provides exact values for multicoloured Ramsey numbers of $P_5$

Enhances understanding of colourings in complete graphs

Abstract

Let $P_t$ denote the path on $t$ vertices. The $r$-coloured Ramsey number of $P_t$, denoted by $R_r(P_t)$, is the minimum integer $n$ such that whenever the complete graph on $n$ vertices is given an $r$-edge-colouring, there exists a monochromatic copy of $P_t$. In this note, we determine $R_r(P_5)$, which is approximately $3r$.