On empty pentagons and hexagons in planar point sets

TL;DR

This paper improves the lower bounds on the minimum number of empty convex pentagons and hexagons in planar point sets, advancing understanding of geometric configurations in combinatorial geometry.

Contribution

It provides new, tighter lower bounds for the number of 5-holes and 6-holes in planar point sets, refining previous results in the field.

Findings

Enhanced lower bounds for 5-holes and 6-holes

Improved combinatorial geometric bounds

Contributes to the understanding of empty convex polygons

Abstract

We give improved lower bounds on the minimum number of $k$-holes (empty convex $k$-gons) in a set of $n$ points in general position in the plane, for $k=5,6$.