Extending Erdős-Beck's theorem to higher dimensions

Abstract Erdős-Beck's theorem states that n points in the plane with at most n − x points collinear define at least cxn lines for some positive constant c. It implies n points in the plane define Θ ( n 2 ) lines unless most of the points (i.e. n − o ( n ) points) are collinear. In this paper, we will present two ways to extend this result to higher dimensions. Given a set S of n points in R d , we want to estimate a lower bound of the number of hyperplanes they define (a hyperplane is defined or spanned by S if it contains d + 1 points of S in general position). Our first result says the number of spanned hyperplanes is at least c x n d − 1 if there exists some hyperplane that contains n − x points of S and is saturated (as defined in Definition 1.3 ). Our second result says n points in R d define Θ ( n d ) hyperplanes unless most of the points belong to the union of a collection of flats whose dimension sum to less than d. Our result has an application to point-hyperplane incidences and a potential application to the point covering problem.