{"acknowledgement":"PP, ZP and MT were partially supported by the Charles University Grant GAUK 421511. ZP was\r\npartially supported by the Charles University Grant SVV-2014-260103. ZP and MT were partially\r\nsupported by the ERC Advanced Grant No. 267165 and by the project CE-ITI (GACR P202/12/G061)\r\nof the Czech Science Foundation. UW was partially supported by the Swiss National Science Foundation\r\n(grants SNSF-200020-138230 and SNSF-PP00P2-138948). Part of this work was done when XG was affiliated with INRIA Nancy Grand-Est and when MT was affiliated with Institutionen för matematik, Kungliga Tekniska Högskolan, then IST Austria.","related_material":{"record":[{"id":"424","status":"public","relation":"later_version"}]},"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 R^d such that the ith reduced Betti number (with Z_2 coefficients in singular homology) of the intersection of any proper subfamily G of F is at most b for every non-negative integer i less or equal to (d-1)/2, then F has Helly number at most h(b,d). These topological conditions are sharp: not controlling any of these 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 from C_*(K) to C_*(R^d). Both techniques are of independent interest."}],"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"},"date_updated":"2024-02-28T12:59:37Z","intvolume":"        34","quality_controlled":"1","day":"01","has_accepted_license":"1","pubrep_id":"501","volume":34,"author":[{"last_name":"Goaoc","first_name":"Xavier","full_name":"Goaoc, Xavier"},{"full_name":"Paták, Pavel","first_name":"Pavel","last_name":"Paták"},{"last_name":"Patakova","orcid":"0000-0002-3975-1683","first_name":"Zuzana","full_name":"Patakova, Zuzana"},{"orcid":"0000-0002-1191-6714","first_name":"Martin","full_name":"Tancer, Martin","last_name":"Tancer"},{"full_name":"Wagner, Uli","orcid":"0000-0002-1494-0568","first_name":"Uli","last_name":"Wagner","id":"36690CA2-F248-11E8-B48F-1D18A9856A87"}],"publist_id":"5665","language":[{"iso":"eng"}],"file_date_updated":"2020-07-14T12:45:00Z","department":[{"_id":"UlWa"}],"page":"507 - 521","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-11T11:52:27Z","alternative_title":["LIPIcs"],"date_published":"2015-01-01T00:00:00Z","article_processing_charge":"No","file":[{"date_updated":"2020-07-14T12:45:00Z","content_type":"application/pdf","file_size":633712,"date_created":"2018-12-12T10:10:09Z","file_name":"IST-2016-501-v1+1_46.pdf","relation":"main_file","file_id":"4794","access_level":"open_access","checksum":"e6881df44d87fe0c2529c9f7b2724614","creator":"system"}],"year":"2015","ddc":["510"],"title":"Bounding Helly numbers via Betti numbers","month":"01","doi":"10.4230/LIPIcs.SOCG.2015.507","_id":"1512","citation":{"short":"X. Goaoc, P. Paták, Z. Patakova, M. Tancer, U. Wagner, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 507–521.","apa":"Goaoc, X., Paták, P., Patakova, Z., Tancer, M., &#38; Wagner, U. (2015). Bounding Helly numbers via Betti numbers (Vol. 34, pp. 507–521). Presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.507\">https://doi.org/10.4230/LIPIcs.SOCG.2015.507</a>","ama":"Goaoc X, Paták P, Patakova Z, Tancer M, Wagner U. Bounding Helly numbers via Betti numbers. In: Vol 34. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2015:507-521. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.507\">10.4230/LIPIcs.SOCG.2015.507</a>","ista":"Goaoc X, Paták P, Patakova Z, Tancer M, Wagner U. 2015. Bounding Helly numbers via Betti numbers. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 34, 507–521.","chicago":"Goaoc, Xavier, Pavel Paták, Zuzana Patakova, Martin Tancer, and Uli Wagner. “Bounding Helly Numbers via Betti Numbers,” 34:507–21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. <a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.507\">https://doi.org/10.4230/LIPIcs.SOCG.2015.507</a>.","ieee":"X. Goaoc, P. Paták, Z. Patakova, M. Tancer, and U. Wagner, “Bounding Helly numbers via Betti numbers,” presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands, 2015, vol. 34, pp. 507–521.","mla":"Goaoc, Xavier, et al. <i>Bounding Helly Numbers via Betti Numbers</i>. Vol. 34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 507–21, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.507\">10.4230/LIPIcs.SOCG.2015.507</a>."},"type":"conference","oa_version":"Submitted Version","scopus_import":"1","oa":1,"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","publication_status":"published","conference":{"name":"SoCG: Symposium on Computational Geometry","end_date":"2015-06-25","start_date":"2015-06-22","location":"Eindhoven, Netherlands"}}