Extremal problems related to Betti numbers of flag complexes

TL;DR

This paper establishes exponential upper bounds on the Betti numbers of flag complexes and related structures, advancing understanding of their topological complexity and providing bounds for various classes of simplicial complexes.

Contribution

It introduces new exponential upper bounds for Betti numbers of flag complexes and independence complexes, extending topological bounds to broader classes of simplicial complexes.

Findings

Betti numbers of any n-vertex flag complex are bounded by 1.32^n

Independence complexes of triangle-free graphs have Betti numbers bounded by 1.25^n

Results imply bounds for neighborhood complexes and related spaces

Abstract

We study the problem of maximizing Betti numbers of simplicial complexes. We prove an upper bound of 1.32^n for the sum of Betti numbers of any n-vertex flag complex and 1.25^n for the independence complex of a triangle-free graph. These findings imply upper bounds for the Betti numbers of various related classes of spaces, including the neighbourhood complex of a graph. We also make some related observations.