Ramsey theory is a central and active branch of combinatorics. Although Ramsey numbers for graphs have been extensively investigated since Ramsey's work in the 1930s, there is still an exponential gap between the best known lower and upper bounds. For $k$-uniform hypergraphs, the bounds are of tower-type, where the height grows with $k$. Here, we give a multidimensional generalisation of Ramsey's Theorem to Cartesian products of graphs, proving that a doubly exponential upper bound suffices in every dimension. More precisely, we prove that for every positive integers $r,n,d$, in any $r$-colouring of the edges of the Cartesian product $\square^{d} K_N$ of $d$ copies of $K_N$, there is a copy of $\square^{d} K_n$ such that the edges in each direction are monochromatic, provided that $N\geq 2^{2^{C_drn^{d}}}$. As an application of our approach we also obtain improvements on the multidimensional Erd\H{o}s-Szekeres Theorem proved by Fishburn and Graham $30$ years ago. Their bound was recently improved by Buci\'c, Sudakov, and Tran, who gave an upper bound that is triply exponential in four or more dimensions. We improve upon their results showing that a doubly expoenential upper bounds holds any number of dimensions.

翻译：Ramsey 理论是一个核心和活跃的组合体分支。 虽然自拉姆齐在1930年代的工作以来,对图表的Ramsey数字进行了广泛调查, 但最知名的下层和上层范围之间仍然存在着指数差距。 对于美元单面高超, 界限是塔型的, 高度以美元增长。 在这里, 我们给 拉姆西 的理论的多维概括化到卡德西语的图表产品, 证明每个维度都有加倍的指数上限。 更确切地说, 我们证明, 每一个正值的整数, $, $, $, $, $, $, $, $, $, 美元, 美元, 美元, 美元, 美元, 最上层, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 数字, 双面, 双面, 美元, 双面, 任何一面, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 越上层, 越上, 越上, 越往下, 越往下, 越往上, 越往下, 越往上, 越往上, 越往, 越往, 越往, 越往, 越往, 越往, 越往, 越往,, 越往, 越往, 越往, 越, 越往, 越往,,,,,, 越往, 越, 越往, 越往, 越往, 越往, 越往, 越往, 越往, 越往, 越往, 越, 越 越 越 越 越 越 越 越 越 越 越 越 越 越 越 越 越往, 越往, 越往, 越往, 越往, 越往, 越往, 越往, 越往, 越往, 越往, 越往, 越往, 越往, 越改进, 越往,