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On the Erdos-Szekeres convex polygon problem (1604.08657v2)
Published 29 Apr 2016 in math.CO and cs.CG
Abstract: Let $ES(n)$ be the smallest integer such that any set of $ES(n)$ points in the plane in general position contains $n$ points in convex position. In their seminal 1935 paper, Erdos and Szekeres showed that $ES(n) \leq {2n - 4\choose n-2} + 1 = 4{n -o(n)}$. In 1960, they showed that $ES(n) \geq 2{n-2} + 1$ and conjectured this to be optimal. In this paper, we nearly settle the Erdos-Szekeres conjecture by showing that $ES(n) =2{n +o(n)}$.