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arXiv · 1510.06255

An improved upper bound for the Erdős-Szekeres conjecture

Hossein Nassajian Mojarrad·Georgios Vlachos
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Abstract

Let $ES(n)$ denote the minimum natural number such that every set of $ES(n)$ points in general position in the plane contains $n$ points in convex position. In 1935, Erdős and Szekeres proved that $ES(n) \le {2n-4 \choose n-2}+1$. In 1961, they obtained the lower bound $2^{n-2}+1 \le ES(n)$, which they conjectured to be optimal. In this paper, we prove that $$ES(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2 \approx \frac{7}{16} {2n-4 \choose n-2}.$$

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Hossein Nassajian Mojarrad, Georgios Vlachos. 2016-05-04. An improved upper bound for the Erdős-Szekeres conjecture. https://arxiv.org/abs/1510.06255

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