  For a set $S$ of (positive definite and integral) quadratic forms with
bounded rank, a quadratic form $f$ is called $S$-universal if it represents all
quadratic forms in $S$. A subset $S_0$ of $S$ is called an $S$-universality
criterion set if any $S_0$-universal quadratic form is $S$-universal. We say
$S_0$ is minimal if there does not exist a proper subset of $S_0$ that is an
$S$-universality criterion set. In this article, we study various properties of
minimal universality criterion sets. In particular, we show that for `most'
binary quadratic forms $f$, minimal $S$-universality criterion sets are unique
in the case when $S$ is the set of all subforms of the binary form $f$.
