On the Number of Almost Empty Monochromatic Triangles
Bhaswar B. Bhattacharya, Sandip Das, Sk Samim Islam, Aashirwad Mohapatra, Ishan Paul, Saumya Sen
TL;DR
This work investigates counting almost empty monochromatic triangles in $c$-colored planar point sets. It establishes two core lower bounds: an $\,\Omega(n^2)$ bound for triangles with at most $c-1$ interior points and an $\,\Omega(n^{4/3})$ bound for triangles with at most $c-2$ interior points, thereby generalizing Pach and Tóth’s results to multiple colors. It further derives the limiting behavior for the number of triangles with interior points in random point sets, showing $\lim_{n\to\infty} \mathbb{E}[Z_{=s}]/n^2 = 2$ and $\lim_{n\to\infty} \mathbb{E}[Z_{\le s}]/n^2 = 2(s+1)$, which imply $\mathbb{E}[X^{(c)}_{\le s}] = \frac{2(s+1)}{c^2} n^2 (1+o(1))$ for random colorings. The proofs combine constructive geometric methods (star triangulations) with a discrepancy-based framework and integral geometry (Blaschke-Petkantschin) to connect combinatorial and probabilistic counts, advancing the theory of colored almost-empty polygons. These results deepen our understanding of how color and interior-point constraints influence the abundance of simplexes in planar point sets, with implications for extremal and probabilistic combinatorial geometry.
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 $Ω(n^2)$ monochromatic triangles with at most $c-1$ interior points and $Ω(n^{\frac{4}{3}})$ 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 Tóth (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.