= Normalized speed of a hereditary hypergraph property
{title2=$d_n=|\mathcal P_n|^{1/\binom nr}$}
For a <hereditary hypergraph property> of $r$-<uniform hypergraphs>, $\mathcal P_n$ is the family on the labelled <vertex set> $[n]$. For $n\geq r$, its normalized speed lies between zero and two and is nonincreasing. Choose a uniformly random member on $n+1$ vertices and record its edge indicators. Each edge occurs in $n+1-r$ vertex-deletion restrictions. The <Shearer inequality> bounds its <information entropy> by the sum of the restriction entropies divided by $n+1-r$. Each restriction has at most $|\mathcal P_n|$ possible values. Division by $\binom{n+1}r$ gives $\log d_{n+1}\leq\log d_n$. Empty families are handled directly by heredity.
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