On Sets of Monochromatic Objects in Bicolored Point Sets

TL;DR

This paper explores the properties and existence of monochromatic geometric objects in bicolored point sets, providing bounds, structural results, and analyzing random configurations for lines, circles, and conics.

Contribution

It offers new bounds on monochromatic lines, structural theorems relating points on conics and cubics, and insights into the behavior of monochromatic objects in random point colorings.

Findings

At least n^2/24 - O(1) monochromatic lines exist under certain conditions.

Red points on a conic with blue points and line constraints imply collinearity of red points.

Expected number of monochromatic lines minimized by near-pencil configurations in random colorings.

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 Mili\'cevi\'c (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.