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On Short Edges in Complete Topological Graphs.

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Title: On Short Edges in Complete Topological Graphs.
Authors: Suk, Andrew1 (AUTHOR) asuk@ucsd.edu
Source: Discrete & Computational Geometry. Jan2026, Vol. 75 Issue 1, p193-204. 12p.
Subjects: Topological graph theory, Complete graphs, Graph theory, Combinatorial geometry, Mathematicians, Matching theory
Abstract: Let h(n) be the minimum integer such that every complete n-vertex simple topological graph contains an edge that crosses at most h(n) other edges. In 2009, Kynčl and Valtr showed that h (n) = O (n 2 / log 1 / 4 n) , and in the other direction, gave constructions showing that h (n) = Ω (n 3 / 2) . In this paper, we prove that h (n) = O (n 7 / 4) . Along the way, we establish a new variant of Chazelle and Welzl's matching theorem for set systems with bounded VC-dimension, which we believe to be of independent interest. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: On Short Edges in Complete Topological Graphs.
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  Data: <searchLink fieldCode="AR" term="%22Suk%2C+Andrew%22">Suk, Andrew</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> asuk@ucsd.edu</i>
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  Data: <searchLink fieldCode="DE" term="%22Topological+graph+theory%22">Topological graph theory</searchLink><br /><searchLink fieldCode="DE" term="%22Complete+graphs%22">Complete graphs</searchLink><br /><searchLink fieldCode="DE" term="%22Graph+theory%22">Graph theory</searchLink><br /><searchLink fieldCode="DE" term="%22Combinatorial+geometry%22">Combinatorial geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematicians%22">Mathematicians</searchLink><br /><searchLink fieldCode="DE" term="%22Matching+theory%22">Matching theory</searchLink>
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  Label: Abstract
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  Data: Let h(n) be the minimum integer such that every complete n-vertex simple topological graph contains an edge that crosses at most h(n) other edges. In 2009, Kynčl and Valtr showed that h (n) = O (n 2 / log 1 / 4 n) , and in the other direction, gave constructions showing that h (n) = Ω (n 3 / 2) . In this paper, we prove that h (n) = O (n 7 / 4) . Along the way, we establish a new variant of Chazelle and Welzl's matching theorem for set systems with bounded VC-dimension, which we believe to be of independent interest. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s00454-024-00692-4
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      – Code: eng
        Text: English
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        PageCount: 12
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    Subjects:
      – SubjectFull: Topological graph theory
        Type: general
      – SubjectFull: Complete graphs
        Type: general
      – SubjectFull: Graph theory
        Type: general
      – SubjectFull: Combinatorial geometry
        Type: general
      – SubjectFull: Mathematicians
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      – SubjectFull: Matching theory
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      – TitleFull: On Short Edges in Complete Topological Graphs.
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              Text: Jan2026
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