# Intersection patterns of planar sets

G. Kalai, Z. Patakova, Discrete and Computational Geometry (2020).

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*Journal Article*|

*Epub ahead of print*|

*English*

**Scopus indexed**

Author

Kalai, Gil;
Patakova, Zuzana

^{IST Austria}^{}Department

Abstract

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.

Publishing Year

Date Published

2020-06-02

Journal Title

Discrete and Computational Geometry

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eISSN

IST-REx-ID

### Cite this

Kalai G, Patakova Z. Intersection patterns of planar sets.

*Discrete and Computational Geometry*. 2020. doi:10.1007/s00454-020-00205-zKalai, G., & Patakova, Z. (2020). Intersection patterns of planar sets.

*Discrete and Computational Geometry*. https://doi.org/10.1007/s00454-020-00205-zKalai, Gil, and Zuzana Patakova. “Intersection Patterns of Planar Sets.”

*Discrete and Computational Geometry*, 2020. https://doi.org/10.1007/s00454-020-00205-z.G. Kalai and Z. Patakova, “Intersection patterns of planar sets,”

*Discrete and Computational Geometry*, 2020.Kalai G, Patakova Z. 2020. Intersection patterns of planar sets. Discrete and Computational Geometry.

Kalai, Gil, and Zuzana Patakova. “Intersection Patterns of Planar Sets.”

*Discrete and Computational Geometry*, Springer Nature, 2020, doi:10.1007/s00454-020-00205-z.**All files available under the following license(s):**

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arXiv 1907.00885