Abstract
In 1935, Erdős and Szekeres proved that every set of n points in general position in the plane contains the vertices of a convex polygon of \(\frac{1}{2}\log _2(n)\) vertices. In 1961, they constructed, for every positive integer t, a set of \(n:=2^{t-2}\) points in general position in the plane, such that every convex polygon with vertices in this set has at most \(\log _2(n)+1\) vertices. In this paper we show how to realize their construction in an integer grid of size \(O(n^2 \log _2(n)^3).\)
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Notes
More accurately, \(P_{r}\) contains a subset with the same order type as \(S_{k,l}.\)
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The research was supported by Conacyt of Mexico Grant 253261.
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Duque, F., Fabila-Monroy, R. & Hidalgo-Toscano, C. Point Sets with Small Integer Coordinates and No Large Convex Polygons. Discrete Comput Geom 59, 461–476 (2018). https://doi.org/10.1007/s00454-017-9931-6
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DOI: https://doi.org/10.1007/s00454-017-9931-6