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A preliminary result for generalized intersecting families

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: $$N^{(u)}_{k,\ell}(s) \; \leq \; \bigg{\lceil} \dfrac{ k - 1 }{\ell - 1} \cdot {s \choose u} \bigg{\rceil}$$