Extending Erdős- Beck's theorem to higher dimensions

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.