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Intersection Patterns of Planar Sets

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Abstract

Let \({\mathcal {A}}=\{A_1,\ldots ,A_n\}\) be a family of sets in the plane. For \(0 \le i < n\), denote by \(f_i\) the number of subsets \(\sigma \) of \(\{1,\ldots ,n\}\) of cardinality \(i+1\) that satisfy \(\bigcap _{i \in \sigma } A_i \ne \emptyset \). Let \(k \ge 2\) be an integer. We prove that if each k-wise and \((k{+}1)\)-wise intersection of sets from \({\mathcal {A}}\) is empty, or a single point, or both open and path-connected, then \(f_{k+1}=0\) implies \(f_k \le cf_{k-1}\) for some positive constant c depending only on k. Similarly, let \(b \ge 2\), \(k > 2b\) be integers. We prove that if each k-wise or \((k{+}1)\)-wise intersection of sets from \({\mathcal {A}}\) has at most b path-connected components, which all are open, then \(f_{k+1}=0\) implies \(f_k \le cf_{k-1}\) for some positive constant c depending only on b and k. These results also extend to two-dimensional compact surfaces.

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Notes

  1. We note that empty set is, by definition, open.

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Acknowledgements

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.

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Correspondence to Gil Kalai.

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Editor in Charge: Kenneth Clarkson

Dedicated to the memory of Branko Grünbaum

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The 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.

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Kalai, G., Patáková, Z. Intersection Patterns of Planar Sets. Discrete Comput Geom 64, 304–323 (2020). https://doi.org/10.1007/s00454-020-00205-z

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