正五边形是典范拉姆齐集
The regular pentagon is canonically Ramsey
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中文总结 AI 辅助
本文证明素边数的正多边形是典范拉姆齐集，该类是此前已知典范拉姆齐集类之外的首批新构型，丰富了典范拉姆齐集的研究成果。
中文摘要 AI 辅助
若存在更大的点集$S\subset \mathbb{R}^{n'}$，使得对$S$的任意着色，要么包含$C$的单色副本，要么包含$C$的彩虹副本，则称点集$C\subset \mathbb{R}^n$是典范拉姆齐集。Mao、Ozeki和Wang引入该概念，证明30-60-90三角形是典范拉姆齐集。此后，诸多其他构型被证实为典范拉姆齐集：作者证明长方体是典范拉姆齐集；Ge、Shu、Xu和Yu证明所有单纯形是典范拉姆齐集；之后作者又证明所有单纯形的乘积是典范拉姆齐集，该类及其子集闭包包含此前所有已知的典范拉姆齐集。本文证明素边数的正多边形是典范拉姆齐集，这是该类之外首批已知的典范拉姆齐集。
英文摘要
A set of points $C\subset \mathbb{R}^n$ is canonically Ramsey if there is some larger set of points $S\subset \mathbb{R}^{n'}$ such that any colouring of $S$ contains either a monochromatic copy of $C$ or a rainbow copy of $C$. Mao, Ozeki, and Wang introduced this notion, showing that the 30-60-90 triangle is canonically Ramsey. Since then, many other configurations have been shown to be canonically Ramsey. The author showed that cuboids are canonically Ramsey. Ge, Shu, Xu, and Yu later showed that all simplices are canonically Ramsey, after which the author showed that all products of simplices are canonically Ramsey, a class which, together with its closure under taking subsets, includes all previously known canonically Ramsey sets. We prove that regular polygons with a prime number of sides are canonically Ramsey---the first known sets outside this class.
