arXiv 2602.176372026-02-19math.COcs.DM

双色点集中的单色对象集合

On Sets of Monochromatic Objects in Bicolored Point Sets

Indian Institute of Technology Bombay（印度理工学院孟买分校）；London School of Economics（伦敦政治经济学院）

Sujoy Bhore, Konrad Swanepoel

备注：: 19 pages, 7 figures

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AI总结：

研究双色点集中的单色直线和圆等几何对象的性质，证明了单色直线的存在性和某些配置下的共线性，以及随机情况下的最小期望单色直线数量。

摘要：

设P是平面上n个点的集合，不全共线，每个点被染成红色或蓝色。经典的Motzkin-Rabin定理保证了存在一条单色直线。受Green和Tao（2013）对Sylvester-Gallai定理开创性工作的启发，我们研究了单色几何对象，如直线、圆和二次曲线的定量和结构性质。我们首先证明，如果不存在超过三个点共线的情况，那么对于足够大的n，至少存在n²/24 - O(1)条单色直线。然后我们证明了Jamison（1986）定理的逆定理：给定n≥6个蓝色点和n个红色点，如果蓝色点位于二次曲线且每条连接两个蓝色点的直线包含一个红色点，则所有红色点共线。我们还解决了Milićević（2018）猜想的最小非平凡情况，证明如果存在5个无三共线的蓝色点和5个红色点，且蓝色点位于二次曲线且每条连接两个蓝色点的直线包含一个红色点，则所有10个点位于三次曲线上。此外，我们分析了随机情况，并证明对于任何非共线的n≥10个独立染色为红色或蓝色的点集，单色直线的期望数量最小化由近于铅笔配置实现。最后，我们研究了单色圆和二次曲线，并展示了若干自然家族中不存在此类单色对象。

英文摘要：

Let $P$ be a set of $n$ points in the plane, not all on a line, each colored \emph{red} or \emph{blue}. The classical Motzkin--Rabin theorem guarantees the existence of a \emph{monochromatic} line. Motivated by the seminal work of Green and Tao (2013) on the Sylvester-Gallai theorem, we investigate the quantitative and structural properties of monochromatic geometric objects, such as lines, circles, and conics. We first show that if no line contains more than three points, then for all sufficiently large $n$ there are at least $n^{2}/24 - O(1)$ monochromatic lines. We then show a converse of a theorem of Jamison (1986): Given $n\ge 6$ blue points and $n$ red points, if the blue points lie on a conic and every line through two blue points contains a red point, then all red points are collinear. We also settle the smallest nontrivial case of a conjecture of Milićević (2018) by showing that if we have $5$ blue points with no three collinear and $5$ red points, if the blue points lie on a conic and every line through two blue points contains a red point, then all $10$ points lie on a cubic curve. Further, we analyze the random setting and show that, for any non-collinear set of $n\ge 10$ points independently colored red or blue, the expected number of monochromatic lines is minimized by the \emph{near-pencil} configuration. Finally, we examine monochromatic circles and conics, and exhibit several natural families in which no such monochromatic objects exist.

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