Core Concepts
The distribution of prime numbers within the "prime grid" (a grid of points in two-dimensional space where both coordinates are prime numbers) does not adhere to any Helly-type theorem. This means that there is no limit to the number of vertices an empty polygon (a polygon with vertices only on prime grid points and no other prime grid points inside) can have.
Dillon, T. (2024). The prime grid contains arbitrarily large empty polygons. arXiv:2411.10549v1 [math.CO].
This research paper aims to prove the conjecture proposed by De Loera, La Haye, Oliveros, and Roldán-Pensado in 2017, stating that the "prime grid" contains empty polygons with an unlimited number of vertices. This implies the absence of a Helly-type theorem for the prime grid.