Maximum Betti Numbers of Čech Complexes

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 Rᵈ R d and any radius satisfies β ₚ = O(Nᵐ) β p = O ( N m ) , with m = min \ p+1, d/2 \ m = min { p + 1 , ⌈ d / 2 ⌉ } . We construct sets in even and odd dimensions that prove this upper bound is asymptotically tight. For example, we describe a set of $$N = 2(n+1)$$ N = 2 ( n + 1 ) points in R³ R 3 and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1 ( n + 1 ) 2 - 1 , and the second Betti number of the Čech complex at the other radius is n² n 2 .