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A Stepping-Up Lemma for Topological Set Systems

Authors Xavier Goaoc, Andreas F. Holmsen, Zuzana Patáková

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Author Details

Xavier Goaoc
  • LORIA, Université de Lorraine, France
Andreas F. Holmsen
  • Department of Mathematical Sciences, KAIST, Daejeon, South Korea
Zuzana Patáková
  • Department of Algebra, Faculty of Mathematics and Physics, Charles University, Czech Republic

Funding

  • Goaoc, Xavier: Institut Universitaire de France.
  • Holmsen, Andreas F.: NRF grant No. 2020R1F1A1A0104849011.
  • Patáková, Zuzana: Charles University projects PRIMUS/21/SCI/014 and UNCE/SCI/022.

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Xavier Goaoc, Andreas F. Holmsen, and Zuzana Patáková. A Stepping-Up Lemma for Topological Set Systems. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.SoCG.2021.40

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Helly-type theorem
  • Topological combinatorics
  • Homological minors
  • Stair convexity
  • Cubical complexes
  • Homological VC dimension
  • Ramsey-type theorem

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