There are many 5-holes
Omar Astudillo-Marb\'an, Oriol Sol\'e-Pi

TL;DR
This paper proves that any set of n points in general position on the plane contains at least on the order of n^{20/11} empty convex pentagons, improving previous bounds and moving closer to the conjectured quadratic lower bound.
Contribution
The authors establish a new lower bound of Ω(n^{20/11}) for the number of empty convex pentagons in point sets, without relying on computer-assisted proofs.
Findings
Improved lower bound of Ω(n^{20/11}) for 5-holes.
The result narrows the gap towards the conjectured Ω(n^2) bound.
Proof does not require computer assistance.
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 empty convex pentagons (also known as 5-holes). This result improves upon the previous bound of obtained by Aicholzer et al. [JCT A, 2020], and significantly narrows the gap with respect to the conjectured 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.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Complexity and Algorithms in Graphs · VLSI and FPGA Design Techniques
