{"PlanS_conform":"1","publisher":"Springer Nature","scopus_import":"1","status":"public","project":[{"call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"has_accepted_license":"1","date_updated":"2026-03-09T11:31:29Z","volume":10,"OA_place":"publisher","arxiv":1,"day":"01","type":"journal_article","intvolume":"        10","department":[{"_id":"HeEd"}],"oa":1,"title":"Maximum persistent Betti numbers of Čech complexes","external_id":{"arxiv":["2409.05241"]},"date_published":"2026-03-01T00:00:00Z","article_number":"5","quality_controlled":"1","article_processing_charge":"Yes (in subscription journal)","article_type":"original","oa_version":"Published Version","author":[{"last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"last_name":"Kahle","full_name":"Kahle, Matthew","first_name":"Matthew"},{"full_name":"Kanazawa, Shu","first_name":"Shu","last_name":"Kanazawa"}],"month":"03","citation":{"mla":"Edelsbrunner, Herbert, et al. “Maximum Persistent Betti Numbers of Čech Complexes.” <i>Journal of Applied and Computational Topology</i>, vol. 10, 5, Springer Nature, 2026, doi:<a href=\"https://doi.org/10.1007/s41468-026-00233-3\">10.1007/s41468-026-00233-3</a>.","ama":"Edelsbrunner H, Kahle M, Kanazawa S. Maximum persistent Betti numbers of Čech complexes. <i>Journal of Applied and Computational Topology</i>. 2026;10. doi:<a href=\"https://doi.org/10.1007/s41468-026-00233-3\">10.1007/s41468-026-00233-3</a>","ieee":"H. Edelsbrunner, M. Kahle, and S. Kanazawa, “Maximum persistent Betti numbers of Čech complexes,” <i>Journal of Applied and Computational Topology</i>, vol. 10. Springer Nature, 2026.","apa":"Edelsbrunner, H., Kahle, M., &#38; Kanazawa, S. (2026). Maximum persistent Betti numbers of Čech complexes. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-026-00233-3\">https://doi.org/10.1007/s41468-026-00233-3</a>","short":"H. Edelsbrunner, M. Kahle, S. Kanazawa, Journal of Applied and Computational Topology 10 (2026).","chicago":"Edelsbrunner, Herbert, Matthew Kahle, and Shu Kanazawa. “Maximum Persistent Betti Numbers of Čech Complexes.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s41468-026-00233-3\">https://doi.org/10.1007/s41468-026-00233-3</a>.","ista":"Edelsbrunner H, Kahle M, Kanazawa S. 2026. Maximum persistent Betti numbers of Čech complexes. Journal of Applied and Computational Topology. 10, 5."},"file_date_updated":"2026-03-09T11:29:30Z","ddc":["500"],"abstract":[{"text":"This note proves that only a linear number of holes in a Cech complex of n points in R^d\r\ncan persist over an interval of constant length. Specifically, for any fixed dimension p <\r\nd and fixed ε > 0, the number of p-dimensional holes in the ˇ Cech complex at radius 1\r\nthat persist to radius 1+ε is bounded above by a constant times n,where n is the number\r\nof points. The proof uses a packing argument supported by relating theCˇ ech complexes\r\nwith corresponding snap complexes over the cells in a partition of space. The argument\r\nis self-contained and elementary, relying on geometric and combinatorial constructions\r\nrather than on the existing theory of sparse approximations or interleavings. The bound\r\nalso applies to Alpha complexes and Vietoris–Rips complexes. While our result can be\r\ninferred from prior work on sparse filtrations, to our knowledge, no explicit statement\r\nor direct proof of this bound appears in the literature.","lang":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"year":"2026","publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"acknowledgement":"The authors would like to thank Michael Lesnick and Primoz Skraba for their helpful comments regarding sparse approximations of filtrations. We are also grateful to the anonymous referees for their careful reading and constructive suggestions. The three authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. I 02979-N35, the U.S. National Science Foundation (NSF-DMS), grant no. 2005630, and a JSPS Grant-in-Aid for Transformative Research Areas (A) (22H05107, Y.H.), EPSRC Research Grant EP/Y008642/1.","language":[{"iso":"eng"}],"doi":"10.1007/s41468-026-00233-3","OA_type":"hybrid","publication_status":"published","date_created":"2026-03-08T23:01:45Z","file":[{"file_size":323111,"creator":"dernst","access_level":"open_access","date_created":"2026-03-09T11:29:30Z","checksum":"0bf6dc430cafa40c08f260fe17d54595","success":1,"file_id":"21416","relation":"main_file","date_updated":"2026-03-09T11:29:30Z","content_type":"application/pdf","file_name":"2026_JourAppliedCompTopology_Edelsbrunner.pdf"}],"_id":"21407","publication":"Journal of Applied and Computational Topology"}