--- _id: '7960' abstract: - lang: eng text: Let A={A1,…,An} be a family of sets in the plane. For 0≤i2b 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 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. Discrete and Computational Geometry. 2020;64:304-323. doi:10.1007/s00454-020-00205-z apa: Kalai, G., & Patakova, Z. (2020). Intersection patterns of planar sets. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00205-z chicago: Kalai, Gil, and Zuzana Patakova. “Intersection Patterns of Planar Sets.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00205-z. ieee: G. Kalai and Z. Patakova, “Intersection patterns of planar sets,” Discrete and Computational Geometry, 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.” Discrete and Computational Geometry, vol. 64, Springer Nature, 2020, pp. 304–23, doi:10.1007/s00454-020-00205-z. 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' ...