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

TL;DR

This paper proves that large point sets in the plane necessarily contain many collinear points or an empty pentagon, advancing understanding in combinatorial geometry and resolving a case of a longstanding conjecture.

Contribution

It generalizes the empty pentagon theorem to include multiple collinear points and addresses an open case of the 'big line or big clique' conjecture.

Findings

Large point sets contain many collinear points or an empty pentagon

Settles an open case of the 'big line or big clique' conjecture

Provides a generalized theorem in combinatorial geometry

Abstract

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 K\'ara, P\'or, and Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005].