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 $\mathrm{\Theta }(n^2)$ 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 ${\mathbb{R}}^d$, 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 $cx{n}^{d-1}$ 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 ${\mathbb{R}}^d$ define $\mathrm{\Theta }(n^d)$ 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.

埃尔德斯-贝克定理指出，平面上n个点最多$ñ-X$共线的点至少为某个正常数c定义cxn条线。这意味着平面中的n个点定义$\mathrm{\Theta }（{ñ}^2）$ 线，除非大多数要点（即 $ñ-Ø（ñ）$ 点）是共线的。

在本文中，我们将提出两种方法将此结果扩展到更高的维度。给定S的n点${\mathbb{[R}}^d$中，我们要估计的下限它们限定超平面的数目（一个超平面由下式定义或跨区的小号如果它包含$d+\mathrm{1个}$的点小号一般位置）。我们的第一个结果说，跨接超平面的数量至少为$CX{ñ}^{d-\mathrm{1个}}$ 如果存在一些包含 $ñ-X$S点和S点饱和（定义1.3中定义）。我们的第二个结果说ñ在点${\mathbb{[R}}^d$ 定义 $\mathrm{\Theta }（{ñ}^d）$除非大多数点都属于平面集合的并集，这些集合的总和小于d。

我们的结果适用于点超平面入射，并且可能适用于点覆盖问题。