@inproceedings{17146,
  abstract     = {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.},
  author       = {Edelsbrunner, Herbert and Pach, János},
  booktitle    = {40th International Symposium on Computational Geometry},
  isbn         = {9783959773164},
  issn         = {1868-8969},
  location     = {Athens, Greece},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Maximum Betti numbers of Čech complexes}},
  doi          = {10.4230/LIPIcs.SoCG.2024.53},
  volume       = {293},
  year         = {2024},
}