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For a positive integer $r$, let $G(r)$ be the smallest $N$ such that,whenever the edges of the Cartesian product $K_N \times K_N$ are $r$-coloured,then there is a rectangle in which both pairs of opposite edges receive thesame colour. In this paper, we improve the upper bounds on $G(r)$ by proving$G(r) \leq \Big(1 - \frac{1}{128}r^{-2}\Big) r^{\binom{r+1}{2}}$, for $r$ largeenough. Unlike the previous improvements, which were based on bounds for thesize of set systems with restricted intersection sizes, our proof is a form ofa quasirandomness argument.

An improved upper bound for the grid Ramsey problem