Flag complexes and homology

We prove several relations on the f -vectors and Betti numbers of flag complexes. For every complex Δ, we show that there exists a balanced with same -vector as whose top-dimensional number is at least thereby extending theorem Frohmader by additionally taking homology into consideration. obtain upper bounds Δ in terms its face numbers. also give quantitative refinement Meshulam establishing lower Δ. This result has continuous analog: If ( d − 1 ) -dimensional -th reduced group dimension ≥ 0 (over some field), then -polynomial satisfies coefficient-wise inequality x + .