Empty Rainbow Triangles in $k$-colored Point Sets

TL;DR

This paper investigates the minimum number of empty rainbow triangles in balanced k-colored point sets in the plane, providing tight bounds and showing the non-existence of empty rainbow quadrilaterals in some cases.

Contribution

It establishes tight asymptotic bounds for the number of empty rainbow triangles and demonstrates the non-existence of empty rainbow quadrilaterals for certain parameters.

Findings

Tight asymptotic bounds for f(k,m)

Existence of empty rainbow triangles in balanced k-colored sets

Non-existence of empty rainbow quadrilaterals for some large k and m

Abstract

Let $S$ be a set of $n$ points in general position in the plane. Suppose that each point of $S$ has been assigned one of $k \ge 3$ possible colors and that there is the same number, $m$, of points of each color class. A polygon with vertices on $S$ is empty if it does not contain points of $S$ in its interior; and it is rainbow if all its vertices have different colors. Let $f(k,m)$ be the minimum number of empty rainbow triangles determined by $S$. In this paper we give tight asymptotic bounds for this function. Furthermore, we show that $S$ may not determine an empty rainbow quadrilateral for some arbitrarily large values of $k$ and $m$.