{"PlanS_conform":"1","title":"Maximum Betti numbers of Čech complexes","abstract":[{"lang":"eng","text":"The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². "}],"author":[{"first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"János","full_name":"Pach, János","id":"E62E3130-B088-11EA-B919-BF823C25FEA4","last_name":"Pach"}],"quality_controlled":"1","oa_version":"Published Version","oa":1,"corr_author":"1","date_updated":"2025-11-20T09:36:13Z","publication":"Discrete & Computational Geometry","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"17146"}]},"scopus_import":"1","language":[{"iso":"eng"}],"status":"public","ddc":["510"],"citation":{"short":"H. Edelsbrunner, J. Pach, Discrete &#38; Computational Geometry (2025).","ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025.","ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. 2025. doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>","ista":"Edelsbrunner H, Pach J. 2025. Maximum Betti numbers of Čech complexes. Discrete &#38; Computational Geometry.","apa":"Edelsbrunner, H., &#38; Pach, J. (2025). Maximum Betti numbers of Čech complexes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-025-00796-5\">https://doi.org/10.1007/s00454-025-00796-5</a>","chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-025-00796-5\">https://doi.org/10.1007/s00454-025-00796-5</a>.","mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>Discrete &#38; Computational Geometry</i>, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00454-025-00796-5\">10.1007/s00454-025-00796-5</a>."},"year":"2025","OA_type":"hybrid","has_accepted_license":"1","main_file_link":[{"url":"https://doi.org/10.1007/s00454-025-00796-5","open_access":"1"}],"department":[{"_id":"HeEd"}],"month":"11","tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"publisher":"Springer Nature","ec_funded":1,"acknowledgement":"The first author is supported by the European Research Council (ERC), grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35. The second author is supported by the European Research Council (ERC), grant “GeoScape” and by the Hungarian Science Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"arxiv":1,"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183"},{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"},{"grant_number":"Z00342","name":"Mathematics, Computer Science","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"publication_status":"epub_ahead","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["2310.14801"]},"_id":"20657","article_processing_charge":"Yes (via OA deal)","OA_place":"publisher","doi":"10.1007/s00454-025-00796-5","day":"10","article_type":"original","date_created":"2025-11-19T09:44:58Z","date_published":"2025-11-10T00:00:00Z","type":"journal_article"}