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Bounding Helly Numbers via Betti Numbers

  • Xavier Goaoc
  • Pavel Paták
  • Zuzana Patáková
  • Martin Tancer
  • Uli Wagner
Chapter

Abstract

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b, d) such that the following holds. If \(\mathcal{F}\) is a finite family of subsets of \(\mathbb{R}^{d}\) such that \(\tilde{\beta }_{i}\left (\bigcap \mathcal{G}\right ) \leq b\) for any \(\mathcal{G} \subsetneq \mathcal{F}\) and every 0 ≤ i ≤ ⌈d∕2⌉− 1 then \(\mathcal{F}\) has Helly number at most h(b, d). Here \(\tilde{\beta }_{i}\) denotes the reduced \(\mathbb{Z}_{2}\)-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these ⌈d∕2⌉ first Betti numbers allow for families with unbounded Helly number.

Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map \(C_{{\ast}}(K) \rightarrow C_{{\ast}}(\mathbb{R}^{d})\).

Notes

Acknowledgements

We would like to express our immense gratitude to Jiří Matoušek, not only for raising the problem addressed in the present paper and valuable discussions about it, but, much more generally, for the privilege of having known him, as our teacher, mentor, collaborator, and friend. Through his tremendous depth and insight, and the generosity with which he shared them, he greatly influenced all of us.

We further thank Jürgen Eckhoff for helpful comments on a preliminary version of the paper, and Andreas Holmsen and Gil Kalai for providing us with useful references.

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Copyright information

© Springer International publishing AG 2017

Authors and Affiliations

  • Xavier Goaoc
    • 1
  • Pavel Paták
    • 2
  • Zuzana Patáková
    • 3
  • Martin Tancer
    • 3
  • Uli Wagner
    • 4
  1. 1.Université Paris-Est Marne-la-ValléeMarne-la-ValléeFrance
  2. 2.Department of AlgebraCharles UniversityPragueCzech Republic
  3. 3.Department of Applied MathematicsCharles UniversityPragueCzech Republic
  4. 4.IST AustriaKlosterneuburgAustria

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