= Shadow ratio for a hereditary set family
{title2=$p_{m+1}/p_m\leq(x_m-m)/(N-m)$}
For a <down-set> on $N$ coordinates, let $p_m=|\mathcal F_m|/\binom Nm$. For $1\leq m<N$ 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 $2m\leq N$, $p_{2m}\leq p_m^2$; 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.
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