On Sets of Monochromatic Objects in Bicolored Point Sets

2026-02-19

AI Summary

This study investigates the existence and structural properties of monochromatic geometric objects—lines, circles, and conics—in red-blue colorings of planar point sets. By integrating the Motzkin–Rabin theorem with Green–Tao’s quantitative framework and leveraging tools from combinatorial geometry, algebraic geometry, and probability, the work establishes asymptotic lower bounds on the number of monochromatic lines under the condition that no four points are collinear. It proves the converse of Jamison’s theorem, verifies the minimal nontrivial case of Milićević’s conjecture, and demonstrates that near-pencil configurations minimize the expected number of monochromatic lines under random colorings. Furthermore, the paper constructs natural families of point sets containing no monochromatic circles or conics and characterizes the structural properties of red-blue point sets lying on a common cubic curve.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Applications of cryptography

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.

Problem

• monochromatic lines
• bicolored point sets
• Sylvester-Gallai theorem
• conics
• Motzkin-Rabin theorem

Innovation

• monochromatic lines
• bicolored point sets
• conic configurations
• Sylvester-Gallai type theorems
• cubic curves