Any finite Euclidean Ramsey set remains Ramsey after adjoining any point outside its affine hull.
A pyramid with a Ramsey base is Ramsey
2 Pith papers cite this work. Polarity classification is still indexing.
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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.
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math.CO 2years
2026 2representative citing papers
citing papers explorer
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One-point extensions of Euclidean Ramsey sets
Any finite Euclidean Ramsey set remains Ramsey after adjoining any point outside its affine hull.
- A nearcircumsphere-Ramsey Theorem for Solvable Transitive Configurations