Finding the smallest integer $N=\mathit{ES}_d(n)$ 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 $\mathit{ES}_2(n)=2^{n−2}+1$ holds, which was nearly settled by Suk in 2016, who showed that $\mathit{ES}_2(n)=2^{n+o(n)}$. We discuss a recent proof that $\mathit{ES}_d(n)=2^{o(n)}$ holds for all d≥3.
Joint work with Dmitrii Zakharov.