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

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 $\frac{1}{6}(10n-k(k+5))$.