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 the same 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 $r$ large enough. Unlike the previous improvements, which were based on bounds for the size of set systems with restricted intersection sizes, our proof is a form of a quasirandomness argument.

中文翻译：

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网格拉姆齐问题的改进上限

对于正整数 $r$，令 $G(r)$ 是最小的 $N$，这样，每当笛卡尔积 $K_N \times K_N$ 的边是 $r$-colored 时，则有一个矩形两对相对的边缘接收相同的颜色。在本文中，我们通过证明 $G(r) \leq \Big(1 - \frac{1}{128}r^{-2}\Big) r^{ 来改进 $G(r)$ 的上界\binom{r+1}{2}}$，对于足够大的 $r$。与以前的改进不同，这些改进基于具有有限交集大小的集合系统大小的界限，我们的证明是一种准随机性论证的形式。