Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Goaoc, Xavier; Holmsen, Andreas F.; Patáková, Zuzana https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-138396
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A Stepping-Up Lemma for Topological Set Systems

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Abstract

Intersection patterns of convex sets in ℝ^d have the remarkable property that for d+1 ≤ k ≤ 𝓁, in any sufficiently large family of convex sets in ℝ^d, if a constant fraction of the k-element subfamilies have nonempty intersection, then a constant fraction of the 𝓁-element subfamilies must also have nonempty intersection. Here, we prove that a similar phenomenon holds for any topological set system ℱ in ℝ^d. Quantitatively, our bounds depend on how complicated the intersection of 𝓁 elements of ℱ can be, as measured by the maximum of the ⌈d/2⌉ first Betti numbers. As an application, we improve the fractional Helly number of set systems with bounded topological complexity due to the third author, from a Ramsey number down to d+1. We also shed some light on a conjecture of Kalai and Meshulam on intersection patterns of sets with bounded homological VC dimension. A key ingredient in our proof is the use of the stair convexity of Bukh, Matoušek and Nivasch to recast a simplicial complex as a homological minor of a cubical complex.

BibTeX - Entry

@InProceedings{goaoc_et_al:LIPIcs.SoCG.2021.40,
  author =	{Goaoc, Xavier and Holmsen, Andreas F. and Pat\'{a}kov\'{a}, Zuzana},
  title =	{{A Stepping-Up Lemma for Topological Set Systems}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{40:1--40:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13839},
  URN =		{urn:nbn:de:0030-drops-138396},
  doi =		{10.4230/LIPIcs.SoCG.2021.40},
  annote =	{Keywords: Helly-type theorem, Topological combinatorics, Homological minors, Stair convexity, Cubical complexes, Homological VC dimension, Ramsey-type theorem}
}

Keywords: Helly-type theorem, Topological combinatorics, Homological minors, Stair convexity, Cubical complexes, Homological VC dimension, Ramsey-type theorem
Seminar: 37th International Symposium on Computational Geometry (SoCG 2021)
Issue date: 2021
Date of publication: 02.06.2021


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