The Erdős-Szekeres problem in three (and higher) dimensions

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, statement which was nearly confirmed in 2016 by Suk, who showed that ES_2(n)=2^{n+o(n)}. In higher dimensions, on the other hand, it has been unclear even what kind of asymptotic behavior to expect from ES_d(n), with conflicting predictions arising over the years. In this talk, we will discuss a recent proof that ES_d(n)=2^{o(n)} holds for all d≥3. Joint work with Dmitrii Zakharov.