A note on the grid Ramsey problem

The grid Ramsey number $G(r)$ is the smallest number $n$ such that every edge-colouring of the grid graph ${\mathrm{\Gamma }}_{n,n}:=K_n\times K_n$ with $r$ colours induces a rectangle whose parallel edges receive the same colour. We show $ G® \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.