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An Improved Upper Bound for the Erdős–Szekeres Conjecture

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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\((n) \le {2n-4 \atopwithdelims ()n-2}+1\). In 1961, they obtained the lower bound \(2^{n-2}+1 \le \mathrm{ES}(n)\), which they conjectured to be optimal. In this paper, we prove that

$$\begin{aligned} \mathrm{ES}(n) \le \Big (\begin{array}{c}{2n-5} \\ {n-2}\end{array}\Big ) -\Big (\begin{array}{c}{2n-8} \\ {n-3}+2\end{array}\Big ) \approx \frac{7}{16} \Big (\begin{array}{c}{2n-4}\\ {n-2}\end{array}\Big ). \end{aligned}$$

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Acknowledgments

Hossein Nassajian Mojarrad was partially supported by Swiss National Science Foundation Grants 200020-144531, 200021-137574, 200020-162884 and 200020-144531.

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Authors and Affiliations

  1. EPFL, Lausanne, Switzerland

    Hossein Nassajian Mojarrad

  2. MIT, Cambridge, USA

    Georgios Vlachos

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  1. Hossein Nassajian Mojarrad
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  2. Georgios Vlachos
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Correspondence to Georgios Vlachos.

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Mojarrad, H.N., Vlachos, G. An Improved Upper Bound for the Erdős–Szekeres Conjecture. Discrete Comput Geom 56, 165–180 (2016). https://doi.org/10.1007/s00454-016-9791-5

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  • DOI: https://doi.org/10.1007/s00454-016-9791-5

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