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 1.32^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.Comment: V4: rewritten, no new result

On the number of minimal transversals in 3-uniform hypergraphs

We prove that the number of minimal transversals (and also the number of maximal independent sets) in a 3-uniform hypergraph with n vertices is at most c n, where c ≈ 1.6702. The best known lower bound for this number, due to Tomescu, is ad n, where d = 10 1 5 ≈ 1.5849 and a is a constant.