Every Large Point Set contains Many Collinear Points or an Empty Pentagon
Zachary Abel, Brad Ballinger, Prosenjit K. Bose, Sébastien Collette, Vida Dujmović, Ferrán Hurtado, Scott Duke Kominers, Stefan Langerman, Attila Pór, David R. Wood · 2009
We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ 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 [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005].