On Sets of Monochromatic Objects in Bicolored Point Sets

TL;DR

This paper advances the quantitative and structural understanding of monochromatic geometric objects in two-colored point sets, extending Motzkin–Rabin and Sylvester–Gallai paradigms. By combining Green–Tao-type structure theorems with a cubic-curve group framework, it proves a tight $n^{2}/24 - O(1)$ lower bound on monochromatic lines when no line contains more than three points and characterizes near-extremal configurations as cosets of finite subgroups of smooth or acnodal cubic curves. It also establishes a converse to Jamison’s result for conics and verifies a smallest nontrivial case of a related conjecture by showing that five blue points on a conic plus five red points force all ten onto a cubic, while in the random coloring setting the near-pencil minimizes the expected number of monochromatic lines for large $n$. Finally, the authors exhibit natural families where no monochromatic circles or conics exist, highlighting limitations and indicating that higher-degree shapes can behave differently from lines in colored incidence geometry.

Abstract

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.

Figures (7)

• Figure 1: Böröczky example: Five blue points on a circle and five red points on a line. There is just one monochromatic line.
• Figure 2: Illustration of Theorem \ref{['thm:2.11']}
• Figure 3: Five blue points on a circle and five non-collinear red points such that the line through any two blue points contains a red point
• Figure 4: Near-pencil: $n$ non-collinear points with exactly $n-1$ on a line
• Figure 5: Five red points not lying on a line or a circle and $\binom{5}{2}$ blue points with no monochromatic circle

Theorems & Definitions (38)

• Theorem 1.1: Sylvester--Gallai theorem
• Theorem 1.2: Motzkin--Rabin theorem
• Example 2.1
• Theorem 2.2
• Lemma 2.1
• proof
• Lemma 2.2: Green--Tao greentao2013
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['thm:1']}