Planar Point Sets Determine Many Pairwise Crossing Segments

04/18/2019

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by   János Pach, et al.

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We show that any set of n points in general position in the plane determines n^1-o(1) pairwise crossing segments. The best previously known lower bound, Ω(√(n)), was proved more than 25 years ago by Aronov, Erdős, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.