= Effective ground-set parameter of a hereditary uniform layer
{title2=$|\mathcal F_m|=\binom{x_m}m$}
For a nonempty degree-$m$ layer of a <down-set> in an $N$-element <Boolean lattice>, the unique real number $x_m\geq m$ with $|\mathcal F_m|=\binom{x_m}m$ measures an effective number of available coordinates. The <Lovász shadow bound> and $\partial\mathcal F_{m+1}\subseteq\mathcal F_m$ imply $x_{m+1}\leq x_m$ whenever the later layer is nonempty. It is a real parameter, not the actual size of the <set family>'s support.
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