book chapter published yes Xavier Goaoc author Pavel Paták author Zuzana Patakova author 0000-0002-3975-1683 Martin Tancer author 0000-0002-1191-6714 Uli Wagner author 36690CA2-F248-11E8-B48F-1D18A9856A870000-0002-1494-0568 MartinLoebl editor JaroslavNešetřil editor RobinThomas editor UlWa department 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). Springer2017 eng 978-331944479-6 1310.461310.1007/978-3-319-44479-6_17 407 - 447 https://research-explorer.ista.ac.at/record/1512 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>. 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. 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> 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>. 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. 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> 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. 4242018-12-11T11:46:24Z2026-06-18T18:48:49Z