On r-Equichromatic Lines with few points in ℂ²

2024/08/27 by Annor, Dickson Y. B.
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.14705

Abstract

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)).

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