可扩展的无四点共圆问题
The extensible no-four-on-a-circle problem
- University of Oxford（牛津大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明了存在密度为Ω(n)的可扩展无四点共圆集合，解决了与Keevash提出的密度问题，方法为加权随机抽样加删除。
AI中文摘要：
我们证明存在一个集合 $S \subset \mathbb{Z}^2$，其中不包含四个点共圆或共线，且当 $n \rightarrow \infty$ 时，$|S \cap [n]^2| = \Omega(n)$。由于 $[n]^2$ 中任何无四点共圆集合的大小为 $O(n)$，这（在常数因子内）解决了当前作者和Keevash提出的关于可扩展无四点共圆构造密度的问题。我们的构造基于从整数格进行加权随机抽样，随后进行仔细删除。
英文摘要：
We show that there exists a set $S \subset \mathbb{Z}^2$ containing no four points on a circle or a line such that $|S \cap [n]^2| = Ω(n)$ as $n \rightarrow \infty$. Since any no-four-on-a-circle set in $[n]^2$ has size $O(n)$, this resolves (up to a constant) a question raised by the current authors and Keevash concerning the density of extensible no-four-on-a-circle constructions. Our construction is based on weighted random sampling from the integer lattice followed by careful deletion.
