A pyramid with a Ramsey base is Ramsey

Abstract

A finite subset $X$ of ${\mathbb R}^d$ is called a Ramsey set if for any number of colours $k$ there exists a dimension $n$ such that whenever ${\mathbb R}^n$ is $k$-coloured there exists a monochromatic congruent copy of $X$. The classification of Ramsey sets is one of the major unsolved problems in the field of Euclidean Ramsey theory. Towards this, Ivan, Leader and Walters recently asked whether adding a point to a Ramsey set outside of its affine hull necessarily produces another Ramsey set. In this note, we answer their question in the affirmative.

Disclosure

“3 is red. These points therefore use at most the remaining r − 1 colours. Since φe is an isometry, they form a congruent copy of C; by (1), they contain a monochromatic copy of X. This completes the proof of Theorem 1.2. Acknowledgements ChatGPT 5.6 (OpenAI) was used during the brainstorming stages of this article. References [1] N. G. de Bruijn and P. Erdős. A colour problem for infinite graphs and a problem in the theory of relations. Nederl. Akad. Wetensch. Proc. Ser. A,”

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Classification

Brainstorming or outlining

Multiplier

2

Verified

2026-08-23T05:07:56.502017Z

Structural counts