Compatible 4-Holes in Point Sets
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Counting interior-disjoint empty convex polygons in a point set is a typical Erdős-Szekeres-type problem. We study this problem for 4-gons. Let P be a set of n points in the plane and in general position. A subset Q of P with four points is called a 4-hole in P if the convex hull of Q is a quadrilateral and does not contain any point of P in its interior. Two 4-holes in P are compatible if their interiors are disjoint. We show that P contains at least 5n/11- 1 pairwise compatible 4-holes. This improves the lower bound of 2(n-2)/5 which is implied by a result of Sakai and Urrutia (2007).
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