{"type":"conference","abstract":[{"lang":"eng","text":"We prove general topological Radon-type theorems for sets in ℝ^d, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly theorem, and consequently the existence of weak ε-nets as well as a (p,q)-theorem. More precisely: Let X be either ℝ^d, smooth real d-manifold, or a finite d-dimensional simplicial complex. Then if F is a finite, intersection-closed family of sets in X such that the ith reduced Betti number (with ℤ₂ coefficients) of any set in F is at most b for every non-negative integer i less or equal to k, then the Radon number of F is bounded in terms of b and X. Here k is the smallest integer larger or equal to d/2 - 1 if X = ℝ^d; k=d-1 if X is a smooth real d-manifold and not a surface, k=0 if X is a surface and k=d if X is a d-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let F be a finite family of open sets in a surface S such that the intersection of any subfamily of F is either empty, or path-connected. Then the fractional Helly number of F is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a (p,q)-theorem for open subsets of a surface."}],"author":[{"full_name":"Patakova, Zuzana","orcid":"0000-0002-3975-1683","id":"48B57058-F248-11E8-B48F-1D18A9856A87","first_name":"Zuzana","last_name":"Patakova"}],"_id":"7989","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file_date_updated":"2020-07-14T12:48:06Z","year":"2020","alternative_title":["LIPIcs"],"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"intvolume":"       164","date_published":"2020-06-01T00:00:00Z","title":"Bounding radon number via Betti numbers","status":"public","doi":"10.4230/LIPIcs.SoCG.2020.61","date_updated":"2025-07-10T11:54:54Z","date_created":"2020-06-22T09:14:18Z","oa":1,"license":"https://creativecommons.org/licenses/by/4.0/","department":[{"_id":"UlWa"}],"publication_status":"published","volume":164,"month":"06","article_processing_charge":"No","file":[{"access_level":"open_access","date_updated":"2020-07-14T12:48:06Z","date_created":"2020-06-23T06:56:23Z","content_type":"application/pdf","relation":"main_file","file_name":"2020_LIPIcsSoCG_Patakova_61.pdf","file_id":"8005","creator":"dernst","file_size":645421,"checksum":"d0996ca5f6eb32ce955ce782b4f2afbe"}],"quality_controlled":"1","publication":"36th International Symposium on Computational Geometry","oa_version":"Published Version","conference":{"start_date":"2020-06-22","location":"Zürich, Switzerland","end_date":"2020-06-26","name":"SoCG: Symposium on Computational Geometry"},"article_number":"61:1-61:13","language":[{"iso":"eng"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","publication_identifier":{"isbn":["9783959771436"],"issn":["1868-8969"]},"arxiv":1,"ddc":["510"],"corr_author":"1","day":"01","has_accepted_license":"1","scopus_import":"1","external_id":{"arxiv":["1908.01677"]},"citation":{"ama":"Patakova Z. Bounding radon number via Betti numbers. In: <i>36th International Symposium on Computational Geometry</i>. Vol 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2020.61\">10.4230/LIPIcs.SoCG.2020.61</a>","apa":"Patakova, Z. (2020). Bounding radon number via Betti numbers. In <i>36th International Symposium on Computational Geometry</i> (Vol. 164). Zürich, Switzerland: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2020.61\">https://doi.org/10.4230/LIPIcs.SoCG.2020.61</a>","short":"Z. Patakova, in:, 36th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.","ieee":"Z. Patakova, “Bounding radon number via Betti numbers,” in <i>36th International Symposium on Computational Geometry</i>, Zürich, Switzerland, 2020, vol. 164.","ista":"Patakova Z. 2020. Bounding radon number via Betti numbers. 36th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 164, 61:1-61:13.","mla":"Patakova, Zuzana. “Bounding Radon Number via Betti Numbers.” <i>36th International Symposium on Computational Geometry</i>, vol. 164, 61:1-61:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2020.61\">10.4230/LIPIcs.SoCG.2020.61</a>.","chicago":"Patakova, Zuzana. “Bounding Radon Number via Betti Numbers.” In <i>36th International Symposium on Computational Geometry</i>, Vol. 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2020.61\">https://doi.org/10.4230/LIPIcs.SoCG.2020.61</a>."}}