Abstract
Let G and R each be a finite set of green and red points, respectively, such that |G|=n, |R|=n−k, G∩R=∅, and the points of G∪R are not all collinear. Let t be the total number of lines determined by G∪R. The number of equichromatic lines (a subset of bichromatic) is at least (t+2n+3−k(k+1))/4. A slightly weaker lower bound exists for bichromatic lines determined by points in ℂ2. For sufficiently large point sets, a proof of a conjecture by Kleitman and Pinchasi is provided. A lower bound of (2t+14n−k(3k+7))/14 is demonstrated for bichromatic lines passing through at most six points. Lower bounds are also established for equichromatic lines passing through at most four, five, or six points.
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Purdy, G.B., Smith, J.W. Bichromatic and Equichromatic Lines in ℂ2 and ℝ2 . Discrete Comput Geom 43, 563–576 (2010). https://doi.org/10.1007/s00454-009-9154-6
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DOI: https://doi.org/10.1007/s00454-009-9154-6