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.

翻译：设$P$为平面上$n$个点的集合（不全部共线），每个点被染为\emph{红色}或\emph{蓝色}。经典的Motzkin--Rabin定理保证了\emph{单色}直线的存在性。受Green和Tao（2013）关于Sylvester-Gallai定理的开创性工作启发，我们研究了单色几何对象（如直线、圆和圆锥曲线）的量化与结构性质。我们首先证明：若不存在包含超过三个点的直线，则对所有充分大的$n$，至少存在$n^{2}/24 - O(1)$条单色直线。随后我们证明了Jamison（1986）定理的一个逆命题：给定$n\ge 6$个蓝点和$n$个红点，若蓝点位于一条圆锥曲线上，且每条通过两个蓝点的直线都包含一个红点，则所有红点共线。我们还解决了Milićević（2018）猜想的最小非平凡情形：若有$5$个蓝点（任意三点不共线）和$5$个红点，且蓝点位于一条圆锥曲线上，同时每条通过两个蓝点的直线都包含一个红点，则所有$10$个点位于一条三次曲线上。此外，我们分析了随机情形并证明：对于任意$n\ge 10$个非共线点的集合独立随机染为红或蓝色，单色直线数量的期望值在\emph{近铅笔型}配置下达到最小。最后，我们考察了单色圆与圆锥曲线，并展示了若干不存在此类单色对象的自然族。