EMPTY PENTAGONS IN POINT SETS WITH COLLINEARITIES
János Barát, Vida Dujmović, Gwenaël Joret, Michael S. Payne, Ludmila Scharf, Daria Schymura, Pavel Valtr, David, R. Wood · 2013
Abstract. 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 328`2 points in the plane contains an empty pentagon or ` collinear points. This is optimal up to a constant factor since the ( ` − 1) × ( ` − 1) grid contains no empty pentagon and no ` collinear points. The previous best known bound was doubly exponential. 1.