Every Large Point Set contains Many Collinear Points or an Empty Pentagon - Abel, Zachary; Ballinger, Brad; Bose, Prosenjit; Collette, Sébastien; Dujmović, Vida; Hurtado, Ferran; Kominers, Scott Duke; Langerman, Stefan; Pór, Attila; Wood, David R. (Graphs and combinatorics, 2011)
We prove the following generalised empty pentagon theorem for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique” conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34(3):497–506, 2005].