Note on the multicolour size-Ramsey number for paths

TL;DR

This paper provides an alternative proof for an upper bound on the size-Ramsey number of paths, showing it is nearly optimal and depends logarithmically on the number of colours.

Contribution

It offers a new proof for Krivelevich's recent bound on the size-Ramsey number of paths, improving understanding of its tightness.

Findings

Upper bound on size-Ramsey number is nearly optimal

The bound depends on the number of colours and path length

Alternative proof technique for existing bounds

Abstract

The size-Ramsey number $\hat{R}(F,r)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with $r$ colours yields a monochromatic copy of $F$. In this short note, we give an alternative proof of the recent result of Krivelevich that $\hat{R}(P_n,r) = O((\log r)r^2 n)$. This upper bound is nearly optimal, since it is also known that $\hat{R}(P_n,r) = \Omega(r^2 n)$.