Two extensions of the Erdis-Szekeres problem

Cited 0 time in webofscience Cited 0 time in scopus
  • Hit : 91
  • Download : 0
According to Suk's breakthrough result on the Erdos-Szekeres problem, any point set in general position in the plane, which has no n elements that form the vertex set of a convex n-gon, has at most 2(n)+ O(n(2/3) log n) points. We strengthen this theorem in two ways. First, we show that the result generalizes to convexity structures induced by pseudoline arrangements. Second, we improve the error term. A family of n convex bodies in the plane is said to be in convex position if the convex hull of the union of no n - 1 of its members contains the remaining one. If any three members are in convex position, we say that the family is in general position. Combining our results with a theorem of Dobbins, Holmsen, and Hubard, we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes Toth and by Pach and Toth, respectively. Let c(n) (and c'(n)) denote the smallest positive integer N with the property that any family of N pairwise disjoint convex bodies in general position (resp., N convex bodies in general position, any pair of which share at most two boundary points) has an n-member subfamily in convex position. We show that c(n) <= c' (n) <= 2(n+O)(root n log n).
Publisher
EUROPEAN MATHEMATICAL SOC
Issue Date
2020-08
Language
English
Article Type
Article
Citation

JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, v.22, no.12, pp.3981 - 3995

ISSN
1435-9855
DOI
10.4171/JEMS/1000
URI
http://hdl.handle.net/10203/279405
Appears in Collection
MA-Journal Papers(저널논문)
Files in This Item
There are no files associated with this item.

qr_code

  • mendeley

    citeulike


rss_1.0 rss_2.0 atom_1.0