 Positive clique error of a truncated lattice meet

Choose a uniform $m$-subset $Z$ and the graph consisting only of its clique. Acceptance by two approximating set families but rejection by their intersection requires two minimal witnesses whose union has size greater than $\ell$. At least one has size greater than $\ell /2$. The sunflower bound for minimal members of a Razborov-closed family and $\mathbb{P}(W\subseteq Z)\le (m/n{)}^{|W|}$ give $2{\sum }_{s>\ell /2}s!(r-1{)}^s(m/n{)}^s\le 4q^{\ell /2}$.