Abstract
We define Π(n) to be the largest number such that for every setP ofn points in the plane, there exist two pointsx, y ε P, where every circle containingx andy contains Π(n) points ofP. We establish lower and upper bounds for Π(n) and show that [n/27]+2≤Π(n)≤[n/4]+1. We define\(\bar \Pi (n)\) for the special case where then points are restricted to be the vertices of a convex polygon. We show that\(\bar \Pi (n) = [n/3] + 1\).
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Hayward, R., Rappaport, D. & Wenger, R. Some extremal results on circles containing points. Discrete Comput Geom 4, 253–258 (1989). https://doi.org/10.1007/BF02187726
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DOI: https://doi.org/10.1007/BF02187726