---
_id: '424'
abstract:
- lang: eng
  text: '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 F is
    a finite family of subsets of Rd such that βi(∩G)≤b for any G⊊F and every 0 ≤
    i ≤ [d/2]-1 then F has Helly number at most h(b, d). Here βi denotes the reduced
    Z2-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*(K)→C*(Rd).'
author:
- first_name: Xavier
  full_name: Goaoc, Xavier
  last_name: Goaoc
- first_name: Pavel
  full_name: Paták, Pavel
  last_name: Paták
- first_name: Zuzana
  full_name: Patakova, Zuzana
  last_name: Patakova
  orcid: 0000-0002-3975-1683
- first_name: Martin
  full_name: Tancer, Martin
  last_name: Tancer
  orcid: 0000-0002-1191-6714
- first_name: Uli
  full_name: Wagner, Uli
  id: 36690CA2-F248-11E8-B48F-1D18A9856A87
  last_name: Wagner
  orcid: 0000-0002-1494-0568
citation:
  ama: 'Goaoc X, Paták P, Patakova Z, Tancer M, Wagner U. Bounding helly numbers via
    betti numbers. In: Loebl M, Nešetřil J, Thomas R, eds. <i>A Journey through Discrete
    Mathematics: A Tribute to Jiri Matousek</i>. A Journey Through Discrete Mathematics.
    Springer; 2017:407-447. doi:<a href="https://doi.org/10.1007/978-3-319-44479-6_17">10.1007/978-3-319-44479-6_17</a>'
  apa: 'Goaoc, X., Paták, P., Patakova, Z., Tancer, M., &#38; Wagner, U. (2017). Bounding
    helly numbers via betti numbers. In M. Loebl, J. Nešetřil, &#38; R. Thomas (Eds.),
    <i>A Journey through Discrete Mathematics: A Tribute to Jiri Matousek</i> (pp.
    407–447). Springer. <a href="https://doi.org/10.1007/978-3-319-44479-6_17">https://doi.org/10.1007/978-3-319-44479-6_17</a>'
  chicago: 'Goaoc, Xavier, Pavel Paták, Zuzana Patakova, Martin Tancer, and Uli Wagner.
    “Bounding Helly Numbers via Betti Numbers.” In <i>A Journey through Discrete Mathematics:
    A Tribute to Jiri Matousek</i>, edited by Martin Loebl, Jaroslav Nešetřil, and
    Robin Thomas, 407–47. A Journey Through Discrete Mathematics. Springer, 2017.
    <a href="https://doi.org/10.1007/978-3-319-44479-6_17">https://doi.org/10.1007/978-3-319-44479-6_17</a>.'
  ieee: 'X. Goaoc, P. Paták, Z. Patakova, M. Tancer, and U. Wagner, “Bounding helly
    numbers via betti numbers,” in <i>A Journey through Discrete Mathematics: A Tribute
    to Jiri Matousek</i>, M. Loebl, J. Nešetřil, and R. Thomas, Eds. Springer, 2017,
    pp. 407–447.'
  ista: 'Goaoc X, Paták P, Patakova Z, Tancer M, Wagner U. 2017.Bounding helly numbers
    via betti numbers. In: A Journey through Discrete Mathematics: A Tribute to Jiri
    Matousek. , 407–447.'
  mla: 'Goaoc, Xavier, et al. “Bounding Helly Numbers via Betti Numbers.” <i>A Journey
    through Discrete Mathematics: A Tribute to Jiri Matousek</i>, edited by Martin
    Loebl et al., Springer, 2017, pp. 407–47, doi:<a href="https://doi.org/10.1007/978-3-319-44479-6_17">10.1007/978-3-319-44479-6_17</a>.'
  short: 'X. Goaoc, P. Paták, Z. Patakova, M. Tancer, U. Wagner, in:, M. Loebl, J.
    Nešetřil, R. Thomas (Eds.), A Journey through Discrete Mathematics: A Tribute
    to Jiri Matousek, Springer, 2017, pp. 407–447.'
date_created: 2018-12-11T11:46:24Z
date_published: 2017-10-06T00:00:00Z
date_updated: 2024-02-28T12:59:37Z
day: '06'
department:
- _id: UlWa
doi: 10.1007/978-3-319-44479-6_17
editor:
- first_name: Martin
  full_name: Loebl, Martin
  last_name: Loebl
- first_name: Jaroslav
  full_name: Nešetřil, Jaroslav
  last_name: Nešetřil
- first_name: Robin
  full_name: Thomas, Robin
  last_name: Thomas
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1310.4613v3
month: '10'
oa: 1
oa_version: Published Version
page: 407 - 447
publication: 'A Journey through Discrete Mathematics: A Tribute to Jiri Matousek'
publication_identifier:
  isbn:
  - 978-331944479-6
publication_status: published
publisher: Springer
publist_id: '7399'
quality_controlled: '1'
related_material:
  record:
  - id: '1512'
    relation: earlier_version
    status: public
scopus_import: 1
series_title: A Journey Through Discrete Mathematics
status: public
title: Bounding helly numbers via betti numbers
type: book_chapter
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
year: '2017'
...