Dmitriy Zakharov: Convex polytopes from fewer points

Let $E{S}_d\left(n\right)$ denote the smallest integer such that any set of $E{S}_d\left(n\right)$ points in $R^d$ in general position contains n points in convex position. In 1960, Erdős and Szekeres showed that $E{S}_2\left(n\right)\ge 2^{n-2}+1$ holds, and famously conjectured that their construction is optimal. This was nearly settled by Suk in 2017, who showed that $E{S}_2\left(n\right)\le 2^{n+o\left(n\right)}$. We show that $E{S}_d\left(n\right)=2^{o\left(n\right)}$holds for all $d\ge 3$. Joint work with Cosmin Pohoata.