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arXiv · 2608.19183

The regular pentagon is canonically Ramsey

Benedict Randall Shaw
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Abstract

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.

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Benedict Randall Shaw. 2026-08-19. The regular pentagon is canonically Ramsey. https://arxiv.org/abs/2608.19183

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