Empty pentagons in point sets with collinearities
  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 328k^2 points in the plane contains an empty pentagon or k collinear points. This is optimal up to a constant factor since the (k-1)x(k-1) grid contains no empty pentagon and no k collinear points. The previous best known bound was doubly exponential. 
J\'anos Bar\'at, Vida Dujmovi\'c, Gwena\"el Joret, Michael S. Payne,   Ludmila Scharf, Daria Schymura, Pavel Valtr and David...
math.CO cs.CG cs.DM