{"quality_controlled":"1","day":"06","status":"public","abstract":[{"lang":"eng","text":"We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b, d) such that the following holds. If F is a finite family of subsets of Rd such that βi(∩G)≤b for any G⊊F and every 0 ≤ i ≤ [d/2]-1 then F has Helly number at most h(b, d). Here βi denotes the reduced Z2-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these [d/2] first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map C*(K)→C*(Rd)."}],"date_updated":"2024-02-28T12:59:37Z","related_material":{"record":[{"status":"public","id":"1512","relation":"earlier_version"}]},"main_file_link":[{"url":"https://arxiv.org/abs/1310.4613v3","open_access":"1"}],"publication_status":"published","series_title":"A Journey Through Discrete Mathematics","scopus_import":1,"oa":1,"publisher":"Springer","citation":{"mla":"Goaoc, Xavier, et al. “Bounding Helly Numbers via Betti Numbers.” <i>A Journey through Discrete Mathematics: A Tribute to Jiri Matousek</i>, edited by Martin Loebl et al., Springer, 2017, pp. 407–47, doi:<a href=\"https://doi.org/10.1007/978-3-319-44479-6_17\">10.1007/978-3-319-44479-6_17</a>.","chicago":"Goaoc, Xavier, Pavel Paták, Zuzana Patakova, Martin Tancer, and Uli Wagner. “Bounding Helly Numbers via Betti Numbers.” In <i>A Journey through Discrete Mathematics: A Tribute to Jiri Matousek</i>, edited by Martin Loebl, Jaroslav Nešetřil, and Robin Thomas, 407–47. A Journey Through Discrete Mathematics. Springer, 2017. <a href=\"https://doi.org/10.1007/978-3-319-44479-6_17\">https://doi.org/10.1007/978-3-319-44479-6_17</a>.","ieee":"X. Goaoc, P. Paták, Z. Patakova, M. Tancer, and U. Wagner, “Bounding helly numbers via betti numbers,” in <i>A Journey through Discrete Mathematics: A Tribute to Jiri Matousek</i>, M. Loebl, J. Nešetřil, and R. Thomas, Eds. Springer, 2017, pp. 407–447.","ama":"Goaoc X, Paták P, Patakova Z, Tancer M, Wagner U. Bounding helly numbers via betti numbers. In: Loebl M, Nešetřil J, Thomas R, eds. <i>A Journey through Discrete Mathematics: A Tribute to Jiri Matousek</i>. A Journey Through Discrete Mathematics. Springer; 2017:407-447. doi:<a href=\"https://doi.org/10.1007/978-3-319-44479-6_17\">10.1007/978-3-319-44479-6_17</a>","ista":"Goaoc X, Paták P, Patakova Z, Tancer M, Wagner U. 2017.Bounding helly numbers via betti numbers. In: A Journey through Discrete Mathematics: A Tribute to Jiri Matousek. , 407–447.","apa":"Goaoc, X., Paták, P., Patakova, Z., Tancer, M., &#38; Wagner, U. (2017). Bounding helly numbers via betti numbers. In M. Loebl, J. Nešetřil, &#38; R. Thomas (Eds.), <i>A Journey through Discrete Mathematics: A Tribute to Jiri Matousek</i> (pp. 407–447). Springer. <a href=\"https://doi.org/10.1007/978-3-319-44479-6_17\">https://doi.org/10.1007/978-3-319-44479-6_17</a>","short":"X. Goaoc, P. Paták, Z. Patakova, M. Tancer, U. Wagner, in:, M. Loebl, J. Nešetřil, R. Thomas (Eds.), A Journey through Discrete Mathematics: A Tribute to Jiri Matousek, Springer, 2017, pp. 407–447."},"type":"book_chapter","oa_version":"Published Version","month":"10","title":"Bounding helly numbers via betti numbers","publication_identifier":{"isbn":["978-331944479-6"]},"doi":"10.1007/978-3-319-44479-6_17","_id":"424","date_published":"2017-10-06T00:00:00Z","year":"2017","date_created":"2018-12-11T11:46:24Z","editor":[{"first_name":"Martin","full_name":"Loebl, Martin","last_name":"Loebl"},{"full_name":"Nešetřil, Jaroslav","first_name":"Jaroslav","last_name":"Nešetřil"},{"first_name":"Robin","full_name":"Thomas, Robin","last_name":"Thomas"}],"publication":"A Journey through Discrete Mathematics: A Tribute to Jiri Matousek","department":[{"_id":"UlWa"}],"page":"407 - 447","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","author":[{"first_name":"Xavier","full_name":"Goaoc, Xavier","last_name":"Goaoc"},{"last_name":"Paták","full_name":"Paták, Pavel","first_name":"Pavel"},{"orcid":"0000-0002-3975-1683","first_name":"Zuzana","full_name":"Patakova, Zuzana","last_name":"Patakova"},{"last_name":"Tancer","orcid":"0000-0002-1191-6714","first_name":"Martin","full_name":"Tancer, Martin"},{"last_name":"Wagner","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli","orcid":"0000-0002-1494-0568","first_name":"Uli"}],"publist_id":"7399","language":[{"iso":"eng"}]}