= Minimal-member bound for a Razborov-closed family
For $r\geq2$, an $r$-closed <set family> has at most $(r-1)^k$ inclusion-minimal members of size $k$. This is stronger than the factorial estimate obtained from the <Erdős–Rado sunflower lemma> alone.
Here is the induction on $r$. For $r=2$, two distinct minimal members force their proper intersection, a contradiction, so at most one minimal member exists. For $r>2$, choose a minimal $k$-<set> $Z$ and group the minimal members $X$ by $W=X\cap Z$. The remainder $X\setminus Z$ is minimal in the family of <sets> $B$ outside $Z$ for which $B\cup W$ belongs to the original family.
That remainder family is $(r-1)$-closed: adding $Z$ as the final witness turns each $(r-1)$-witness forcing step into an original $r$-witness step. Induction bounds the group for $W$ by $(r-2)^{k-|W|}$; the group $W=Z$ contains only $Z$. Therefore the <binomial theorem> gives
$$
\sum_{W\subseteq Z}(r-2)^{k-|W|}=(r-1)^k.
$$
The case $k=0$ has at most the single empty <set>. The forcing argument explains why the stronger bound uses the full closure rule rather than only a sunflower prohibition.<Razborov's Method of Approximations by Timothy Gowers>[Lemma 4.1]{ref}
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