 Witness-pair concentration for an intersecting uniform family

For an intersecting family of $r$-sets, some element $x$ is avoided by at most $(r-1{)}^2(\genfrac{}{}{0ex}{}{n-2}{r-2})$ members. If two members meet exactly at $x$, every member avoiding $x$ contains one element from each of their disjoint remainders. If no such pair exists, apply the pair-cover bound for a two-intersecting uniform family.