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 3 lines passing through at most four points is at least 1/6 (10n − k(k + 5)).