{"department":[{"_id":"UlWa"}],"oa_version":"Published Version","status":"public","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","publication_identifier":{"isbn":["9783959771436"],"issn":["18688969"]},"ddc":["510"],"year":"2020","abstract":[{"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.","lang":"eng"}],"title":"Bounding radon number via Betti numbers","date_updated":"2021-01-12T08:16:22Z","month":"06","scopus_import":"1","article_processing_charge":"No","has_accepted_license":"1","date_created":"2020-06-22T09:14:18Z","article_number":"61:1-61:13","type":"conference","citation":{"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>.","short":"Z. Patakova, in:, 36th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.","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.","ieee":"Z. Patakova, “Bounding radon number via Betti numbers,” in <i>36th International Symposium on Computational Geometry</i>, Zürich, Switzerland, 2020, vol. 164.","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>","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>."},"file":[{"file_id":"8005","checksum":"d0996ca5f6eb32ce955ce782b4f2afbe","date_created":"2020-06-23T06:56:23Z","file_name":"2020_LIPIcsSoCG_Patakova_61.pdf","relation":"main_file","access_level":"open_access","file_size":645421,"date_updated":"2020-07-14T12:48:06Z","content_type":"application/pdf","creator":"dernst"}],"file_date_updated":"2020-07-14T12:48:06Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","language":[{"iso":"eng"}],"intvolume":"       164","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"oa":1,"volume":164,"_id":"7989","alternative_title":["LIPIcs"],"doi":"10.4230/LIPIcs.SoCG.2020.61","author":[{"first_name":"Zuzana","id":"48B57058-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-3975-1683","last_name":"Patakova","full_name":"Patakova, Zuzana"}],"external_id":{"arxiv":["1908.01677"]},"publication":"36th International Symposium on Computational Geometry","day":"01","publication_status":"published","date_published":"2020-06-01T00:00:00Z","conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2020-06-22","end_date":"2020-06-26","location":"Zürich, Switzerland"}}