# Maximum Betti Numbers of Čech Complexes

## TL;DR

This paper shows that the maximum Betti numbers of Čech complexes in different dimensions match the theoretical upper bounds.

## Contribution

The paper proves the asymptotic tightness of the upper bound for Betti numbers of Čech complexes in both even and odd dimensions.

## Key findings

- The upper bound for Betti numbers in Čech complexes is asymptotically tight in even and odd dimensions.
- A specific example in 3D space demonstrates the tightness of the bound for the first and second Betti numbers.

## Abstract

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 \documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb R}}^d$$\end{document}Rd and any radius satisfies \documentclass[12pt]{minimal}
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\begin{document}$${\beta }_{p}{} = O(N^{m})$$\end{document}βp=O(Nm), with \documentclass[12pt]{minimal}
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\begin{document}$$m = \min \{ p+1, {\big \lceil d/2 \big \rceil } \}$$\end{document}m=min{p+1,⌈d/2⌉}. We construct sets in even and odd dimensions that prove this upper bound is asymptotically tight. For example, we describe a set of \documentclass[12pt]{minimal}
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\begin{document}$$N = 2(n+1)$$\end{document}N=2(n+1) points in \documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb R}}^3$$\end{document}R3 and two radii such that the first Betti number of the Čech complex at one radius is \documentclass[12pt]{minimal}
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\begin{document}$$(n+1)^2 - 1$$\end{document}(n+1)2-1, and the second Betti number of the Čech complex at the other radius is \documentclass[12pt]{minimal}
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\begin{document}$$n^2$$\end{document}n2.