Convex polytopes from fewer points

Abstract

Let $ES_{d}(n)$ be the smallest integer such that any set of $ES_{d}(n)$ points in $\mathbb{R}^{d}$ in general position contains $n$ points in convex position. In 1960, Erd\H{o}s and Szekeres showed that $ES_{2}(n) \geq 2^{n-2} + 1$ holds, and famously conjectured that their construction is optimal. This was nearly settled by Suk in 2017, who showed that $ES_{2}(n) \leq 2^{n+o(n)}$. In this paper, we prove that $$ES_{d}(n) = 2^{o(n)}$$ holds for all $d \geq 3$. In particular, this establishes that, in higher dimensions, substantially fewer points are needed in order to ensure the presence of a convex polytope on $n$ vertices, compared to how many are required in the plane.