Effective ground-set parameter of a hereditary uniform layer ($|{\mathcal{F}}_m|=\left(\genfrac{}{}{0ex}{}{x_m}{m}\right)$)

For a nonempty degree-$m$ layer of a down-set in an $N$-element Boolean lattice, the unique real number $x_m\ge m$ with $|{\mathcal{F}}_m|=\left(\genfrac{}{}{0ex}{}{x_m}{m}\right)$ measures an effective number of available coordinates. The Lovász shadow bound and $\partial {\mathcal{F}}_{m+1}\subseteq {\mathcal{F}}_m$ imply $x_{m+1}\le x_m$ whenever the later layer is nonempty. It is a real parameter, not the actual size of the set family's support.

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.