Empty pentagons in point sets with collinearities

TL;DR

This paper proves that any sufficiently large finite point set in the plane must contain either an empty pentagon or k collinear points, improving bounds from doubly exponential to quadratic in k.

Contribution

It establishes a nearly optimal quadratic bound on the size of point sets guaranteeing an empty pentagon or k collinear points, advancing combinatorial geometry understanding.

Findings

Every set with at least 328k^2 points contains an empty pentagon or k collinear points.

The bound is tight up to a constant factor, as grids show.

Improves previous bounds from doubly exponential to quadratic in k.

Abstract

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.