The Erdős-Szekeres problem for non-crossing convex sets (Q2874619)
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scientific article; zbMATH DE number 6327864
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The Erdős-Szekeres problem for non-crossing convex sets |
scientific article; zbMATH DE number 6327864 |
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8 August 2014
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pseudoline
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pseudoline arrangement
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CC-system
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uniform rank 3 acyclic oriented matroid
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generalized configuration
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Erdős-Szekeres theorem
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non-crossing convex bodies
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happy ending theorem
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0.9468254
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0.92680854
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0.9268085
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0.9232659
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0.9224541
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0.9157467
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0.9138348
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0.9127358
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The Erdős-Szekeres problem for non-crossing convex sets (English)
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According to a result of \textit{J. E. Goodman} and \textit{R. Pollack} [Congr. Numerantium 32, 383--394 (1981; Zbl 0495.05012)] for every integer \(n\geq 3\) there exists a smallest positive integer \(g(n)\) such that any generalized configuration of size \(g(n)\), in which every triple is convexly independent, contains a convexly independent subset of size \(n\). According to a result of \textit{J. Pach} and \textit{G. Tóth} [Geom. Dedicata 81, No. 1--3, 1--12 (2000; Zbl 0959.52002)] for every integer \(n\geq 3\) there exists a smallest positive integer \(h_1(n)\) such that for every family of \(h_1(n) \) non-crossing bodies in the Euclidean plane, in which every triple is convexly independent, contains a convexly independent subfamily of size \(n\).NEWLINENEWLINENEWLINE The paper under review shows that \(g(n)=h_1(n)\).
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