Maximum persistent Betti numbers of Čech complexes

This note proves that only a linear number of holes in a Čech complex of n points in $R^d$ can persist over an interval of constant length. Specifically, for any fixed dimension $p<d$ and fixed   $\epsilon >0$ , the number of p-dimensional holes in the Čech complex at radius 1 that persist to radius $1+\epsilon$ is bounded above by a constant times n, where n is the number of points. The proof uses a packing argument supported by relating the Čech complexes with corresponding snap complexes over the cells in a partition of space. The argument is self-contained and elementary, relying on geometric and combinatorial constructions rather than on the existing theory of sparse approximations or interleavings. The bound also applies to Alpha complexes and Vietoris-Rips complexes. While our result can be inferred from prior work on sparse filtrations, to our knowledge, no explicit statement or direct proof of this bound appears in the literature.