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

我们证明了标志复合物的f向量和Betti数的几种关系。对于每个标志复数Δ，我们表明存在一个与Δ相同的f-矢量的平衡复数，并且其顶维Betti数至少为Δ的复数，从而通过另外考虑同源性来扩展Frohmader定理。我们根据其面数来获得Δ的顶级维Betti数的上限。我们还通过建立Δ的f维矢量的下界（根据Δ的维Betti数），对Meshulam定理进行了定量细化。此结果具有连续的模拟量：如果Δ为a$（d-\mathrm{1个}）$维标志复合体，其 $（d-\mathrm{1个}）$-第一个简化的同源群具有维数 $\mathrm{一种}\ge 0$（在某个域上），则Δ的f多项式满足系数式不等式$F_{\mathrm{\Delta }}（X）\ge {（\mathrm{1个}+（\sqrt[d]{\mathrm{一种}}+\mathrm{1个}）X）}^d$。