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 $R^d$ and any radius satisfies ${\beta }_p=O\left(N^m\right)$ , with $m=min\{p+1,\lceil d/2\rceil \}$ . We construct sets in even and odd dimensions that prove this upper bound is asymptotically tight. For example, we describe a set of $N=2(n+1)$ points in $R^3$ 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 $n^2$ .