There are many 5-holes

Abstract

Given a set P of points on the plane, a polygon with vertices in P is said to be empty if it contains no element of P in its interior. We show that every set of n points in general position on the plane determines at least $\hat{I}\text{c◯}(n^{20/11})$ empty convex pentagons (also known as 5-holes). This result improves upon the previous bound of $\hat{I}\text{c◯}(n\cdot (\log n{)}^{4/5})$ obtained by Aicholzer et al. [JCT A, 2020], and significantly narrows the gap with respect to the conjectured $\hat{I}\text{c◯}(n^2)$ lower bound (which, if true, would be tight). Unlike some of the other works in this line of research, our proof does not require computer assistance.

17 pages, 11 figures