欧氏拉姆齐集的单点扩张
One-point extensions of Euclidean Ramsey sets
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中文总结 AI 辅助
该研究证明有限欧氏拉姆齐集X的仿射包外添加任意点仍为欧氏拉姆齐集，解答了Ivan等人的猜想，给出了分情形的严格证明。
中文摘要 AI 辅助
设X为有限欧氏拉姆齐集，我们证明：在X的仿射包之外添加任意一点，可得到另一个欧氏拉姆齐集，从而解答了Ivan、Leader和Walters提出的猜想。我们先在附加假设（新点的正交投影属于conv(X)，且其与aff(X)的距离足够大）下给出基础的乘积证明，随后结合E-拉姆齐构型的乘积定理、Kříž的轨道粘合定理与循环构造，证明一般情形，且无需对X施加传递性假设。
英文摘要
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 Kř\'ıž's orbit-gluing theorem with a cyclic construction. No transitivity assumption on $X$ is needed.
