Extending Erd\H{o}s- Beck's theorem to higher dimensions

TL;DR

This paper extends the Erdős-Beck theorem from the plane to higher dimensions, providing new bounds on incidences and potential applications to covering problems.

Contribution

It introduces two methods to generalize the Erdős-Beck theorem to higher dimensions, advancing understanding of point-hyperplane incidences.

Findings

Extended bounds for point-hyperplane incidences in higher dimensions

Proposed two methods for higher-dimensional generalization

Potential applications to point covering problems

Abstract

Erd\H{o}s-Beck theorem states that $n$ points in the plane with at most $n-x$ points collinear define at least $c xn$ lines for some positive constant $c$. In this paper, we will present two ways to extend this result to higher dimensions. Our result has application to point-hyperplane incidences and potential application to the point covering problem.