On the Number of Almost Empty Monochromatic Triangles

2026-01-26

AI Summary

This study investigates lower bounds on the number of “almost empty” monochromatic triangles—triangles whose vertices are of the same color and whose interiors contain at most a few points of other colors—in a multicolored point set in general position in the plane. By combining combinatorial geometry and probabilistic methods, the authors extend the classical result of Pach and Tóth on empty monochromatic triangles in two-colored point sets to arbitrary $c$-colorings and more general settings allowing up to $c-1$ or $c-2$ interior points. The main contributions include proving that any $c$-coloring yields $\Omega(n^2)$ monochromatic triangles with at most $c-1$ interior points and $\Omega(n^{4/3})$ such triangles with at most $c-2$ interior points. Additionally, the work provides the first asymptotic estimate for the expected number of monochromatic triangles containing exactly $s$ interior points under random point sets and random colorings.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

In this paper, we consider the problem of counting almost empty monochromatic triangles in colored planar point sets, that is, triangles whose vertices are all assigned the same color and that contain only a few interior points. Specifically, we show that any $c$-coloring of a set of $n$ points in the plane in general position (that is, no three on a line) contains $\Omega(n^2)$ monochromatic triangles with at most $c-1$ interior points and $\Omega(n^{\frac{4}{3}})$ monochromatic triangles with at most $c-2$ interior points, for any fixed $c \geq 2$. The latter, in particular, generalizes the result of Pach and T\'{o}th (2013) on the number of monochromatic empty triangles in 2-colored point sets, to the setting of multiple colors and monochromatic triangles with a few interior points. We also derive the limiting value of the expected number of triangles with $s$ interior points in random point sets, for any integer $s \geq 0$. As a result, we obtain the expected number of monochromatic triangles with at most $s$ interior points in random colorings of random point sets.

Problem

• monochromatic triangles
• almost empty triangles
• colored point sets
• interior points
• combinatorial geometry

Innovation

• monochromatic triangles
• almost empty triangles
• colored point sets
• combinatorial geometry
• random point sets