Finding the smallest integer N=ES_d(n) such that in every configuration of N points in 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 ES_2(n)=2^{n−2}+1 holds, which was nearly settled by Suk in 2016, who showed that ES_2(n)≤2^{n+o(n)}. We discuss a recent proof that ES_d(n)=2^{o(n)} holds for all d≥3.

Joint work with Dmitrii Zakharov.