Finding the smallest integer N=ESd(n) such that in every configuration of N points in ℝd 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 ES2(n)=2n−2+1 holds, statement which was nearly confirmed in 2016 by Suk, who showed that ES2(n)=2n+o(n). In higher dimensions, on the other hand, it has been unclear even what kind of asymptotic behaviour to expect from ESd(n), with conflicting predictions arising over the years. In this talk, we will discuss a recent proof that ESd(n)=2o(n) holds for all d≥3. Joint work with Dmitrii Zakharov.
This video is part of the Institute for Advanced Study‘s Special year research seminar.
