conference paper 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 UlWa department SoCG: Symposium on Computational Geometry 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. https://research-explorer.ista.ac.at/download/1512/4794/IST-2016-501-v1+1_46.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Schloss Dagstuhl - Leibniz-Zentrum für Informatik2015Eindhoven, Netherlands eng 10.4230/LIPIcs.SOCG.2015.507 34507 - 521 https://research-explorer.ista.ac.at/record/424 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> 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. 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. 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>. 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>. 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> X. Goaoc, P. Paták, Z. Patakova, M. Tancer, U. Wagner, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 507–521. 15122018-12-11T11:52:27Z2026-06-18T18:48:48Z