AI 中文总结
研究单纯形的乘积是否为典范拉姆齐，通过扩展前人成果，证明所有单纯形的乘积都是典范拉姆齐的。
AI 中文摘要
如果存在某个点集\(S\subset \mathbb{R}^{n'}\)，使得对\(S\)用任意数量的颜色进行任何着色，都必定包含\(C\)的单色副本或\(C\)的彩虹副本，那么点集\(C \subset \mathbb{R}^n\)就被称为典范拉姆齐的。Mao、Ozeki和Wang引入了这个概念，证明了30 - 60 - 90三角形是典范拉姆齐的。此后，又确定了各种其他典范拉姆齐构型。作者证明了长方体是典范拉姆齐的，而Ge、Shu、Xu和Yu最近证明了单纯形是典范拉姆齐的。本文扩展了这两个结果，证明了所有单纯形的乘积都是典范拉姆齐的。
英文摘要
A set of points $C \subset \mathbb{R}^n$ is called canonically Ramsey if there is some set of points $S\subset \mathbb{R}^{n'}$ such that any colouring of $S$, using any number of colours, must contain either a monochromatic copy of $C$ or a rainbow copy of $C$. Mao, Ozeki, and Wang introduced this notion, showing that 30-60-90 triangles are canonically Ramsey. Since then, various other canonically Ramsey configurations have been identified. The author showed that cuboids are canonically Ramsey, while Ge, Shu, Xu, and Yu recently showed that simplices are canonically Ramsey. We extend both of these results, proving that all products of simplices are canonically Ramsey.
Comments9 pages; minor edits and a revised argument for Lemma 6, results unchanged
