Bounding Helly numbers via Betti numbers LIPIcs Goaoc, Xavier Paták, Pavel Patakova, Zuzana ; https://orcid.org/0000-0002-3975-1683 Tancer, Martin ; https://orcid.org/0000-0002-1191-6714 Wagner, Uli ; https://orcid.org/0000-0002-1494-0568 ddc:510 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. Schloss Dagstuhl - Leibniz-Zentrum für Informatik 2015 info:eu-repo/semantics/conferenceObject doc-type:conferenceObject text http://purl.org/coar/resource_type/c_5794 https://research-explorer.ista.ac.at/record/1512 https://research-explorer.ista.ac.at/download/1512/4794 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> eng info:eu-repo/semantics/altIdentifier/doi/10.4230/LIPIcs.SOCG.2015.507 info:eu-repo/semantics/openAccess