  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.

Combinatorics