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Empty pentagons in point sets with collinearities

par Barát, Janos;Dujmović, Vida V.;Joret, Gwenaël ;Payne, Michael ;Scharf, Ludmila;Schymura, Daria;Valtr, Pavel;Wood, David
Référence SIAM journal on discrete mathematics, 29, 1, page (198-209)
Publication Publié, 2015

Article révisé par les pairs

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Résumé :

An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328ℓ2 points in the plane contains an empty pentagon or ℓ collinear points. This is optimal up to a constant factor since the (ℓ - 1) x (ℓ - 1) square lattice contains no empty pentagon and no ℓcollinear points. The previous best known bound was doubly exponential.