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On the Erdős-Szekeres convex polygon problem. (English) Zbl 1370.52032
P. Erdős and G. Szekeres [Compos. Math. 2, 463–470 (1935; Zbl 0012.27010)] showed that for every positive integer $$n$$ there is a least positive integer $$\mathrm{ES}(n)$$ such that among any $$ES(n)$$ points in the plane, such that no 3 are collinear, some $$n$$ points are the vertices of a convex $$n$$-gon. They gave two proofs, one of them was Ramsey theoretic, while the other was a geometric argument, using cups and caps, providing $$\mathrm{ES}(n)\leq \binom{2n-4}{n-2}$$. Later they returned to this problem: P. Erdős and G. Szekeres [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 3–4, 53–62 (1961; Zbl 0103.15502)] showed $$\mathrm{ES}(n)\geq 2^{n-2}+1$$ and made a conjecture that $$\mathrm{ES}(n)= 2^{n-2}+1$$. A number of improvements have been made on the upper bound, but they at most gained a constant multiplicative factor.
The paper under review shows $$\mathrm{ES}(n)=2^{n+o(n)}$$, a breakthrough result, providing the logarithmic asymptotics for $$\mathrm{ES}(n)$$. The result hinges on some results of A. Pór and P. Valtr [Discrete Comput. Geom. 28, No. 4, 625–637 (2002; Zbl 1019.52011)] regarding cups and caps. The error term in the exponent was originally $$6n^{2/3}\log n$$, which was reduced by Gábor Tardos to $$O\sqrt{n\log n})$$.

MSC:
 52C10 Erdős problems and related topics of discrete geometry 05D10 Ramsey theory
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