Extremal problems related to Betti numbers of flag complexes

We study the problem of maximizing Betti numbers of simplicial complexes. We prove an upper bound of 1.32n1.32n for the sum of Betti numbers of any nn-vertex flag complex and 1.25n1.25n 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.