{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7885","name":"Bounding Helly Numbers via Betti Numbers","abstract":"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.\r\n\r\nOur 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.","keywords":["Helly-type theorem","Ramsey\u2019s theorem","Embedding of simplicial complexes","Homological almost-embedding","Betti numbers"],"author":[{"@type":"Person","name":"Goaoc, Xavier","givenName":"Xavier","familyName":"Goaoc"},{"@type":"Person","name":"Pat\u00e1k, Pavel","givenName":"Pavel","familyName":"Pat\u00e1k"},{"@type":"Person","name":"Pat\u00e1kov\u00e1, Zuzana","givenName":"Zuzana","familyName":"Pat\u00e1kov\u00e1"},{"@type":"Person","name":"Tancer, Martin","givenName":"Martin","familyName":"Tancer"},{"@type":"Person","name":"Wagner, Uli","givenName":"Uli","familyName":"Wagner"}],"position":40,"pageStart":507,"pageEnd":521,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Goaoc, Xavier","givenName":"Xavier","familyName":"Goaoc"},{"@type":"Person","name":"Pat\u00e1k, Pavel","givenName":"Pavel","familyName":"Pat\u00e1k"},{"@type":"Person","name":"Pat\u00e1kov\u00e1, Zuzana","givenName":"Zuzana","familyName":"Pat\u00e1kov\u00e1"},{"@type":"Person","name":"Tancer, Martin","givenName":"Martin","familyName":"Tancer"},{"@type":"Person","name":"Wagner, Uli","givenName":"Uli","familyName":"Wagner"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.507","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/1101.6006","http:\/\/arxiv.org\/abs\/1310.4613"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6237","volumeNumber":34,"name":"31st International Symposium on Computational Geometry (SoCG 2015)","dateCreated":"2015-06-12","datePublished":"2015-06-12","editor":[{"@type":"Person","name":"Arge, Lars","givenName":"Lars","familyName":"Arge"},{"@type":"Person","name":"Pach, J\u00e1nos","givenName":"J\u00e1nos","familyName":"Pach"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7885","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6237"}}}