On the Number of Almost Empty Monochromatic Triangles

TL;DR

This paper investigates the minimum number of almost empty monochromatic triangles in multi-colored planar point sets, providing bounds and extending previous results to multiple colors and interior points.

Contribution

It generalizes prior work on empty triangles to multiple colors and interior points, establishing new lower bounds and expected values for such triangles.

Findings

Any c-coloring contains Ω(n^2) triangles with at most c-1 interior points.

Any c-coloring contains Ω(n^{4/3}) triangles with at most c-2 interior points.

Derived the expected number of triangles with s interior points in random point sets.

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.