Every Large Point Set contains Many Collinear Points or an Empty Pentagon

Abstract

We prove the following generalised empty pentagon theorem: for every integer $\ell \ge 2$, 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].

References in corpus (1)

• Drawings of Planar Graphs with Few Slopes and Segments

Cited by in corpus (4)

• Flip Distance Between Triangulations of a Simple Polygon is NP-Complete
• Flip Distance Between Triangulations of a Planar Point Set is APX-Hard
• Empty pentagons in point sets with collinearities
• Blocking Coloured Point Sets