{"year":"2017","abstract":[{"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).","lang":"eng"}],"oa":1,"author":[{"first_name":"Xavier","last_name":"Goaoc","full_name":"Goaoc, Xavier"},{"first_name":"Pavel","last_name":"Paták","full_name":"Paták, Pavel"},{"full_name":"Patakova, Zuzana","first_name":"Zuzana","orcid":"0000-0002-3975-1683","last_name":"Patakova"},{"orcid":"0000-0002-1191-6714","last_name":"Tancer","first_name":"Martin","full_name":"Tancer, Martin"},{"full_name":"Wagner, Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli","orcid":"0000-0002-1494-0568","last_name":"Wagner"}],"language":[{"iso":"eng"}],"day":"06","editor":[{"full_name":"Loebl, Martin","last_name":"Loebl","first_name":"Martin"},{"full_name":"Nešetřil, Jaroslav","last_name":"Nešetřil","first_name":"Jaroslav"},{"first_name":"Robin","last_name":"Thomas","full_name":"Thomas, Robin"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1310.4613v3"}],"page":"407 - 447","publication_identifier":{"isbn":["978-331944479-6"]},"citation":{"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.","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>.","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.","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>","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>","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."},"related_material":{"record":[{"relation":"earlier_version","status":"public","id":"1512"}]},"title":"Bounding helly numbers via betti numbers","date_published":"2017-10-06T00:00:00Z","quality_controlled":"1","scopus_import":1,"publication_status":"published","month":"10","date_created":"2018-12-11T11:46:24Z","publisher":"Springer","publication":"A Journey through Discrete Mathematics: A Tribute to Jiri Matousek","date_updated":"2024-02-28T12:59:37Z","type":"book_chapter","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","publist_id":"7399","doi":"10.1007/978-3-319-44479-6_17","department":[{"_id":"UlWa"}],"oa_version":"Published Version","series_title":"A Journey Through Discrete Mathematics","status":"public","_id":"424"}