The multicolour size Ramsey number of a path

TL;DR

This paper determines the multicolour size Ramsey number of paths, establishing tight bounds for fixed numbers of colours and path lengths, and improves the lower bound for these numbers.

Contribution

It provides the first tight bounds for the multicolour size Ramsey number of paths and enhances the lower bound for these numbers.

Findings

For fixed r ≥ 2 and k ≥ 100 log r, the size Ramsey number is Θ((r^2 log r) k).

Established tight bounds for the multicolour size Ramsey number of paths.

Improved the lower bound on the size Ramsey number of paths.

Abstract

In this paper, we determine the $r$-colour size Ramsey number of the path $P_k$, up to constants. In particular, for every fixed $r \geq 2$ and $k \geq 100\log r$, we have \[ \widehat{R}_r(P_k)=\Theta((r^2 \log r) \, k).\] Perhaps surprisingly, we do this by improving the lower bound on $\widehat{R}_r(P_k)$.