Maximum Persistent Betti Numbers of \v{C}ech Complexes

This note proves that only a linear number of holes in a \v{C}ech complex of $n$ points in $\mathbb{R}^d$ can persist over an interval of constant length. Specifically, for any fixed dimension $p < d$ and fixed $\varepsilon > 0$, the number of $p$-dimensional holes in the \v{C}ech complex at radius $1$ that persist to radius $1 + \varepsilon$ 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 \v{C}ech 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.