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].

Big line or big clique conjecture, Empty hexagon, Empty pentagon, Empty quadrilateral, Erdo{double acute}s-Szekeres theorem, Happy end problem
Graphs and Combinatorics
School of Computer Science

Abel, Z. (Zachary), Ballinger, B. (Brad), Bose, P, Collette, S. (Sébastien), Dujmović, V, Hurtado, F. (Ferran), … Wood, D. (2011). Every Large Point Set contains Many Collinear Points or an Empty Pentagon. Graphs and Combinatorics, 27(1), 47–60. doi:10.1007/s00373-010-0957-2