How many hyperplanes does it take to separate $n$ points in $\mathbb{R}^m$?
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 $P(2,n)=\lceil n/2\rceil$.