  Intersecting families and blocking sets feature prominently in extremal
combinatorics. We examine the following generalization of an intersecting
family investigated by Hajnal, Rothschild, and others. If $s \geq 1$, $k \geq
2$, and $u \geq 1$ are integers, then say that an $s$-uniform family
$\mathcal{F}$ is $(k,u)$-intersecting if for all $A_1, A_2, \cdots, A_k \in
\mathcal{F}$, $|A_i \cap A_j| \geq u$ for some $1 \leq i < j \leq k$. In this
note, we investigate the following parameter. If $s$, $k$, $u$, $\ell$ are
integers satisfying $s \geq 1$, $k \geq 2$, $1 \leq u \leq s$, and $2 \leq \ell
< k$, then let $N^{(u)}_{k,\ell}(s)$ denote the smallest integer $r$, if it
exists, such that any $(k,u)$-intersecting $s$-uniform family is the union of
at most $r$ families that are $(\ell,u)$-intersecting. Using a Sunflower Lemma
type argument, we prove that $N^{(u)}_{k,\ell}(s)$ always exists and that the
following inequality always holds: $$N^{(u)}_{k,\ell}(s) \; \leq \;
\bigg{\lceil} \dfrac{ k - 1 }{\ell - 1} \cdot {s \choose u} \bigg{\rceil}$$
