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Abstract: 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 Erd?s-Szekeres problem. In 1935, Erd?s 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.