---
_id: '7960'
abstract:
- lang: eng
  text: Let A={A1,…,An} be a family of sets in the plane. For 0≤i<n, denote by fi
    the number of subsets σ of {1,…,n} of cardinality i+1 that satisfy ⋂i∈σAi≠∅. Let
    k≥2 be an integer. We prove that if each k-wise and (k+1)-wise intersection of
    sets from A is empty, or a single point, or both open and path-connected, then
    fk+1=0 implies fk≤cfk−1 for some positive constant c depending only on k. Similarly,
    let b≥2, k>2b be integers. We prove that if each k-wise or (k+1)-wise intersection
    of sets from A has at most b path-connected components, which all are open, then
    fk+1=0 implies fk≤cfk−1 for some positive constant c depending only on b and k.
    These results also extend to two-dimensional compact surfaces.
acknowledgement: "We are very grateful to Pavel Paták for many helpful discussions
  and remarks. We also thank the referees for helpful comments, which greatly improved
  the presentation.\r\nThe project was supported by ERC Advanced Grant 320924. GK
  was also partially supported by NSF grant DMS1300120. The research stay of ZP at
  IST Austria is funded by the project CZ.02.2.69/0.0/0.0/17_050/0008466 Improvement
  of internationalization in the field of research and development at Charles University,
  through the support of quality projects MSCA-IF."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Gil
  full_name: Kalai, Gil
  last_name: Kalai
- first_name: Zuzana
  full_name: Patakova, Zuzana
  id: 48B57058-F248-11E8-B48F-1D18A9856A87
  last_name: Patakova
  orcid: 0000-0002-3975-1683
citation:
  ama: Kalai G, Patakova Z. Intersection patterns of planar sets. <i>Discrete and
    Computational Geometry</i>. 2020;64:304-323. doi:<a href="https://doi.org/10.1007/s00454-020-00205-z">10.1007/s00454-020-00205-z</a>
  apa: Kalai, G., &#38; Patakova, Z. (2020). Intersection patterns of planar sets.
    <i>Discrete and Computational Geometry</i>. Springer Nature. <a href="https://doi.org/10.1007/s00454-020-00205-z">https://doi.org/10.1007/s00454-020-00205-z</a>
  chicago: Kalai, Gil, and Zuzana Patakova. “Intersection Patterns of Planar Sets.”
    <i>Discrete and Computational Geometry</i>. Springer Nature, 2020. <a href="https://doi.org/10.1007/s00454-020-00205-z">https://doi.org/10.1007/s00454-020-00205-z</a>.
  ieee: G. Kalai and Z. Patakova, “Intersection patterns of planar sets,” <i>Discrete
    and Computational Geometry</i>, vol. 64. Springer Nature, pp. 304–323, 2020.
  ista: Kalai G, Patakova Z. 2020. Intersection patterns of planar sets. Discrete
    and Computational Geometry. 64, 304–323.
  mla: Kalai, Gil, and Zuzana Patakova. “Intersection Patterns of Planar Sets.” <i>Discrete
    and Computational Geometry</i>, vol. 64, Springer Nature, 2020, pp. 304–23, doi:<a
    href="https://doi.org/10.1007/s00454-020-00205-z">10.1007/s00454-020-00205-z</a>.
  short: G. Kalai, Z. Patakova, Discrete and Computational Geometry 64 (2020) 304–323.
date_created: 2020-06-14T22:00:50Z
date_published: 2020-09-01T00:00:00Z
date_updated: 2023-08-21T08:26:34Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s00454-020-00205-z
external_id:
  arxiv:
  - '1907.00885'
  isi:
  - '000537329400001'
intvolume: '        64'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1907.00885
month: '09'
oa: 1
oa_version: Preprint
page: 304-323
publication: Discrete and Computational Geometry
publication_identifier:
  eissn:
  - '14320444'
  issn:
  - '01795376'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Intersection patterns of planar sets
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 64
year: '2020'
...