Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons
SIAM Journal on Discrete Mathematics
Computational Geometry: Theory and Applications
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
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For a point set P on the plane, a four element subset S ⊂ P is called a 4-hole of P if the convex hull of S is a quadrilateral and contains no point of P in its interior. Let R be a point set on the plane. We say that a point set B covers all the 4-holes of R if any 4-hole of R contains an element of B in its interior. We show that if |R|≥ 2|B| + 5 then B cannot cover all the 4-holes of R. A similar result is shown for a point set R in convex position. We also show a point set R for which any point set B that covers all the 4-holes of R has approximately 2|R| points.