How many hyperplanes does it take to separate $n$ points in $\mathbb{R}^m$?

Question

By 'separate', I mean that each point lies in its own little region/cell.

For instance, it takes a minimum of $P = 4$ lines to separate $n = 7$ points in $\mathbb{R}^2$ ($m=2$), assuming that no 3 points lie on a single line (i.e. are in general position):

(Regular heptagon)

Now, in general, at least how many hyperplanes $P(m, n)$ does it take to separate $n$ points in $\mathbb{R}^m$ (assuming general position)?

Answer 1

This question was considered by Ralph P. Boland and Jorge Urrutia in the paper “Separating Collections of Points in Euclidean Spaces”. I don't read this paper yet. As I understood, the authors showed that $$\lceil (n-1)/m\rceil\le P(m,n)\le \lceil(n-2^{\lceil\log m\rceil})/m\rceil+\lceil\log m\rceil,$$ and $P(2,n)=\lceil n/2\rceil$.