A note on the grid Ramsey problem

TL;DR

This paper investigates the grid Ramsey number, providing a new upper bound that improves upon previous results by Gyárfás, through combinatorial analysis of edge-coloured grid graphs.

Contribution

It introduces a refined upper bound for the grid Ramsey number G(r), advancing the understanding of edge-colouring properties in grid graphs.

Findings

New upper bound for G(r) improves previous estimates

Refined combinatorial techniques used in the proof

Results contribute to Ramsey theory and graph colouring literature

Abstract

The grid Ramsey number $ G(r) $ is the smallest number $ n $ such that every edge-colouring of the grid graph $\Gamma_{n,n} := K_n \times K_n$ with $r$ colours induces a rectangle whose parallel edges receive the same colour. We show $ G(r) \leq r^{\binom{r+1}{2}} - \left( 1/4 - o(1) \right) r^{\binom{r}{2}+1} $, slightly improving the currently best known upper bound due to Gy\'arf\'as.