On $r$-Equichromatic Lines with few points in $\mathbb{C}^2$

TL;DR

This paper establishes lower bounds on the number of r-equichromatic lines with few points in the complex plane, revealing combinatorial properties of point sets with restricted collinearity.

Contribution

It provides new lower bounds for 1- and 2-equichromatic lines with limited points, advancing understanding of geometric configurations in complex spaces.

Findings

Lower bounds for 1-equichromatic lines passing through at most six points.

Lower bounds for 2-equichromatic lines passing through at most four points.

Results depend on maximum collinearity constraints of the point set.

Abstract

Let $P$ be a set of $n$ green and $n - k$ red points in $\mathbb{C}^2$. A line determined by $i$ green and $j$ red points such that $i + j \ge 2$ and $|i - j| \le r$ is called \emph{r-equichromatic}. We establish lower bounds for $1$-equichromatic and $2$-equichromatic lines. In particular, we show that if at most $2n-k-2$ points of $P$ are collinear, then the number of $1$-equichromatic lines passing through at most six points is at least $\frac{1}{4}(6n-k(k+3))$, and if at most $\frac{2}{3}(2n - k)$ points of $P$ are collinear, then the number of $2$-equichromatic lines passing through at most four points is at least $\frac{1}{6}(10n - k(k + 5))$.