{"intvolume":"       164","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","article_number":"61:1-61:13","title":"Bounding radon number via Betti numbers","date_created":"2020-06-22T09:14:18Z","publication":"36th International Symposium on Computational Geometry","author":[{"orcid":"0000-0002-3975-1683","first_name":"Zuzana","full_name":"Patakova, Zuzana","last_name":"Patakova","id":"48B57058-F248-11E8-B48F-1D18A9856A87"}],"status":"public","scopus_import":"1","file":[{"file_name":"2020_LIPIcsSoCG_Patakova_61.pdf","access_level":"open_access","date_created":"2020-06-23T06:56:23Z","file_size":645421,"relation":"main_file","content_type":"application/pdf","date_updated":"2020-07-14T12:48:06Z","checksum":"d0996ca5f6eb32ce955ce782b4f2afbe","creator":"dernst","file_id":"8005"}],"_id":"7989","language":[{"iso":"eng"}],"doi":"10.4230/LIPIcs.SoCG.2020.61","ddc":["510"],"alternative_title":["LIPIcs"],"external_id":{"arxiv":["1908.01677"]},"corr_author":"1","file_date_updated":"2020-07-14T12:48:06Z","conference":{"name":"SoCG: Symposium on Computational Geometry","end_date":"2020-06-26","location":"Zürich, Switzerland","start_date":"2020-06-22"},"article_processing_charge":"No","year":"2020","publication_identifier":{"issn":["1868-8969"],"isbn":["9783959771436"]},"volume":164,"type":"conference","tmp":{"image":"/images/cc_by.png","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)"},"date_published":"2020-06-01T00:00:00Z","department":[{"_id":"UlWa"}],"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"}],"has_accepted_license":"1","oa_version":"Published Version","day":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"06","publication_status":"published","arxiv":1,"oa":1,"quality_controlled":"1","date_updated":"2025-04-10T11:11:49Z"}