Extending Erdös- Beck's theorem to higher dimensions

Thao Do

Published 2016-06-30Version 1

Erd\"os-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 $c$. It roughly implies that $n$ points in the plane span $\Theta(n^2)$ lines unless most of the points (i.e. $n-o(n)$ points) are collinear. In this paper we will extend this result to higher dimensions: $n$ points in $\mathbb{R}^d$ define $\Theta(n^d)$ hyperplanes unless most of the points belong to the union of a collection of flats whose sum of dimension is strictly less than $d$.