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 $\mathbb{R}^d$ 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 Čech 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 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.

💡 Research Summary

This paper investigates the maximal number of persistent topological features that can appear in a Čech complex built on (n) points in (\mathbb{R}^d). For a fixed homological dimension (p<d) and a fixed positive tolerance (\varepsilon), the authors consider the number of (p)-dimensional holes that exist at scale (r=1) and survive until scale (r=1+\varepsilon). They denote this quantity by (M_{p,d,\varepsilon}(n)) and prove that it grows at most linearly in (n); more precisely, there exists a constant (C=C(p,d,\varepsilon)) such that (M_{p,d,\varepsilon}(n)\le C,n) for all (n).

The motivation comes from earlier work (Edelsbrunner and Păch, 2024) showing that the unrestricted maximal Betti number (M_{p,d}(n)) can be as large as (n^{\min{p+1,\lceil d/2\rceil}}). Those constructions create many short-lived holes that disappear after a tiny increase in the radius. By fixing a positive interval (