# Maximum persistent Betti numbers of Čech complexes

## TL;DR

This paper shows that the number of persistent holes in Čech complexes is linear with respect to the number of points in a fixed-dimensional space.

## Contribution

The paper provides a direct, elementary proof that the number of persistent p-dimensional holes in Čech complexes is linear in n.

## Key findings

- The number of p-dimensional holes in Čech complexes that persist over a constant interval is linear in n.
- The result also applies to Alpha and Vietoris–Rips complexes.
- The proof uses geometric and combinatorial methods rather than sparse approximation theory.

## Abstract

This note proves that only a linear number of holes in a Čech complex of n points in \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbb {R}^d$$\end{document}Rd can persist over an interval of constant length. Specifically, for any fixed dimension \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p<d$$\end{document}p<d and fixed  \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\varepsilon >0$$\end{document}ε>0, the number of p-dimensional holes in the Čech complex at radius 1 that persist to radius \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$1+\varepsilon $$\end{document}1+ε 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.