Flag complexes and homology.

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