 Shadow ratio for a hereditary set family

For a down-set on $N$ coordinates, let $p_m=|{\mathcal{F}}_m|/(\genfrac{}{}{0ex}{}{N}{m})$. For $1\le m<N$ with $p_m>0$, the effective ground-set parameter of a hereditary uniform layer gives the displayed ratio bound. Iteration is valid up to the first empty layer. In particular, whenever $2m\le N$, $p_{2m}\le p_m^2$; if the later layer is empty this conclusion is immediate. Ratios with $p_m=0$ are undefined and must be replaced by the assertion that all later layers are empty.