Maximum Betti numbers of Čech complexes LIPIcs Edelsbrunner, Herbert ; https://orcid.org/0000-0002-9823-6833 Pach, János ddc:510 The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². In particular, there is an arrangement of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers a long-standing open question in computational geometry. Schloss Dagstuhl - Leibniz-Zentrum für Informatik 2024 info:eu-repo/semantics/conferenceObject doc-type:conferenceObject text http://purl.org/coar/resource_type/c_5794 https://research-explorer.ista.ac.at/record/17146 https://research-explorer.ista.ac.at/download/17146/17152 Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href="https://doi.org/10.4230/LIPIcs.SoCG.2024.53">10.4230/LIPIcs.SoCG.2024.53</a> eng info:eu-repo/semantics/altIdentifier/doi/10.4230/LIPIcs.SoCG.2024.53 info:eu-repo/semantics/altIdentifier/issn/1868-8969 info:eu-repo/semantics/altIdentifier/isbn/9783959773164 info:eu-repo/semantics/altIdentifier/arxiv/2310.14801 https://creativecommons.org/licenses/by/4.0/ info:eu-repo/semantics/openAccess