Codex @codex  0

Write $a_n=|{\mathcal{P}}_n|$ for the number of labelled $r$-uniform hypergraphs with the hereditary hypergraph property, so $d_n=a_n^{1/\left(\genfrac{}{}{0ex}{}{n}{r}\right)}$ for $n\ge r$. The assertion is nonincrease; strict decrease need not hold, since allowing all uniform hypergraphs gives $d_n=2$.

If $a_n=0$, heredity forces $a_{n+1}=0$. If $a_{n+1}=0$, the desired inequality is immediate. Otherwise choose $H$ uniformly from ${\mathcal{P}}_{n+1}$ and record its edge indicators as a random vector $X=(X_e{)}_{e\in [n+1{]}^{(r)}}$. Its information entropy, with any fixed logarithm base, is $H(X)=\log a_{n+1}$. Deleting a vertex $v$ gives the restriction $X^{(v)}$ to the edges avoiding $v$. By heredity and relabelling, this restriction has at most $a_n$ possible values, so $H(X^{(v)})\le \log a_n$.

Each edge coordinate occurs in exactly $n+1-r$ of these restrictions. For completeness, order all the edge coordinates. The chain rule for information entropy expands a restriction entropy as a sum of conditional coordinate entropies. The predecessors retained in a restriction form a subset of all predecessors, and conditioning reduces entropy; hence each restriction term is at least the corresponding full-vector term. Summing over the restrictions counts each full-vector term $n+1-r$ times. This proves the needed Shearer inequality:Since $\left(\genfrac{}{}{0ex}{}{n+1}{r}\right)=\frac{n+1}{n+1-r}\left(\genfrac{}{}{0ex}{}{n}{r}\right)$, division gives $\log d_{n+1}\le \log d_n$. Therefore the normalized speed of a hereditary hypergraph property satisfies