Publication at Faculty of Mathematics and Physics |
2015
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 328l^2 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 l collinear points. The previous best known bound was doubly exponential.