On problems related to crossing families
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Given a set of points in the plane, a crossing family is a collection of segments, each joining two of the points, such that every two segments intersect internally. Aronov et al. [Combinatorica, 14(2):127-134, 1994] proved that any set of n points contains a crossing family of size Ω(√(n)). They also mentioned that there exist point sets whose maximum crossing family uses at most n/2 of the points. We improve the upper bound on the size of crossing families to 5n/24. We also introduce a few generalizations of crossing families, and give several lower and upper bounds on our generalized notions.
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