On r-Equichromatic Lines with few points in C²

Let P be a set of n green and $n - k$ red points in C². A line determined by i green and j red points such that i + j ≥ 2 and |i - j| ≤ r is called 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 1/4(6n-k(k+3)), and if at most 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 1/6(10n - k(k + 5)).