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 $\Gamma_{n,n} := K_n \times K_n$ 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.