A preliminary result for generalized intersecting families

TL;DR

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Contribution

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Abstract

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