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) x (? - 1) square lattice contains no empty pentagon and no ?collinear points. The previous best known bound was doubly exponential.
- Erd˝os problems
- discrete geometry