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arXiv · 2608.11736

One-point extensions of Euclidean Ramsey sets

Mostafa Mirabi
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Abstract

Let $X$ be a finite Euclidean Ramsey set. We prove that adjoining any point outside the affine hull of $X$ gives another Euclidean Ramsey set, answering a conjecture of Ivan, Leader, and Walters. We first give an elementary product proof under the additional assumptions that the orthogonal projection of the new point lies in $conv(X)$ and that its distance from $aff(X)$ is sufficiently large. We then prove the general case by combining the product theorem for $E$-Ramsey configurations and K\v{r}\'{\i}\v{z}'s orbit-gluing theorem with a cyclic construction. No transitivity assumption on $X$ is needed.

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Mostafa Mirabi. 2026-08-12. One-point extensions of Euclidean Ramsey sets. https://arxiv.org/abs/2608.11736

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