Cosmin Pohoata, Convex polytopes from fewer points

Finding the smallest integer $N=E{S}_d(n)$ such that in every configuration of $N$ points in ${\mathbb{R}}^d$ in general position, there exist $n$ points in convex position is one of the most classical problems in extremal combinatorics, known as the Erdős-Szekeres problem. In 1935, Erdős and Szekeres famously conjectured that $E{S}_2(n)=2^{n-2}+1$ holds, which was nearly settled by Suk in 2016, who showed that $E{S}_2(n)\le 2^{n+o(n)}$. We discuss a recent proof that $E{S}_d(n)=2^{o(n)}$ holds for all $d\ge 3$. Joint work with Dmitrii Zakharov.