An improved upper bound for the Erd\H{o}s-Szekeres conjecture

Hossein Nassajian Mojarrad; Georgios Vlachos

An improved upper bound for the Erd\H{o}s-Szekeres conjecture

Combinatorics 2016-05-05 v2

Authors: Hossein Nassajian Mojarrad , Georgios Vlachos

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, Erd\H{o}s 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 $ES(n) \le {2n-5 \choose n-2}-{2n-8 \choose n-3}+2 \approx \frac{7}{16} {2n-4 \choose n-2}.$

Keywords

Erdős-Szekeres conjecture upper bound convex position general position combinatorial geometry extremal set theory point sets

Cite

@article{arxiv.1510.06255,
  title  = {An improved upper bound for the Erd\H{o}s-Szekeres conjecture},
  author = {Hossein Nassajian Mojarrad and Georgios Vlachos},
  journal= {arXiv preprint arXiv:1510.06255},
  year   = {2016}
}

Comments

15 pages, 10 figures. Accepted to Discrete & Computational Geometry (DCG). This second version is a result of the merger of the first version (arXiv:1510.06255v1) and the paper (arXiv:1505.07549)

An improved upper bound for the Erd\H{o}s-Szekeres conjecture

Abstract

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