Big convex polytopes or rich hyperplanes

Abstract

For natural numbers $n$ and $l>d\ge 2$, let $E{S}_d(l,n)$ be the minimum $N$ such that any set of at least $N$ points in ${\mathbb{R}}^d$ contains either $l$ points contained in a common $(d-1)$-dimensional hyperplane or $n$ points in convex position. In this paper, we give the upper and lower bounds for $E{S}_d(l,n)$.

6 pages, 3 figures