An inequality for the number of independent sets of matroids with an application to the forest-tree ratio of graphs

Let $M=(E,\mathcal{I})$ be a matroid of rank $r$. Let ${\mathcal{I}}_k$ be the independent sets of size $k$, and let $I_k=\mathrm{\mid }{\mathcal{I}}_k\mathrm{\mid }$ and $I=\mathrm{\mid }\mathcal{I}\mathrm{\mid }$. We show that if every set $F\in {\mathcal{I}}_{r-1}$ is contained in at least $\delta$ bases, then $$\ln \left(\frac{I}{I_r}\right)⩾\frac{I_{r-1}}{I_r}\cdot \delta \ln (1+\frac{1}{\delta })\mathrm{.}$$ In particular, we have $$\frac{I}{I_r}⩾2^{I_{r-1}\mathrm{/}{I}_r}\mathrm{.}$$ By combining this result with several other ideas, we prove that if $G$ is a simple connected graph on $n$ vertices, and $F(G)$ and $T(G)$ denote its numbers of spanning forests and spanning trees, respectively, then $$\frac{F(G)}{T(G)}⩾\frac{F(K_n)}{T(K_n)},$$ where $K_n$ is the complete graph on $n$ vertices. Equality holds if and only if $G=K_n$.