Generalization of Beck's theorem

Question

Beck's theorem is a classical result in discrete geometry which describes about the geometry of points in the plane. The result states that a finite collections of points in the plane fall into one of two extremes; one where a large fraction of points lie on a single line, and one where a large number of lines are needed to connect all the points.

Formally, the statement of the theorem is as follows:

Theorem: Let $S$ be a set of $n$ points in the plane. If at most $(n-k)$ points lie on any line for some $0 \leq k < n-2$, then there exist $\Omega(nk)$ lines determined by the points of $S$.

Now, my question is that if the generalization/variation of the above result is known in the case of higher dimension $(dim\geq 3)$?

Answer 1

If you are asking if the same statement holds true or not in higher dimensions, it does. Just project all the points to a random 2-dimensional plane.

Another natural generalization is to consider hyperplanes instead of lines. Here, you have Beck's "other" theorem:

Theorem: For any $d \geq 2$, there are constants $\beta_d, \gamma_d \in (0,1/2]$ such that for any set $P$ of $n$ points in $\mathbb{R}^d$, at least one of the following holds:

• a hyperplane contains at least $\beta_d n$ points of $P$
• the $d$-tuples of $P$ span at least $\gamma_d n^d$ distinct hyperplanes.

Here, a hyperplane is a $(d-1)$-dimensional flat. For $d=2$, this is essentially what you stated. The restriction that $\beta_d \leq 1/2$ is from the example of having two skew lines in $\mathbb{R}^3$ containing $n/2$ points each, where the number of spanned planes is $n$. For more, see: http://www.cs.elte.hu/~elekes/Abstracts/elest.ps