Intersecting Families in a Subset of Boolean Lattices

Jun Wang, Huajun Zhang


Let $n, r$ and $\ell$ be distinct positive integers with $r < \ell\leq n/2$, and let $X_1$ and $X_2$ be two disjoint sets with the same size $n$. Define $$\mathcal{F}=\left\{A\in \binom{X}{r+\ell}: \mbox{$|A\cap X_1|=r$ or $\ell$}\right\},$$ where $X=X_1\cup X_2$. In this paper, we prove that if $\mathcal{S}$ is an intersecting family in $\mathcal{F}$, then $|\mathcal{S}|\leq \binom{n-1}{r-1}\binom{n}{\ell}+\binom{n-1}{\ell-1}\binom{n}{r}$, and equality holds if and only if $\mathcal{S}=\{A\in\mathcal{F}: a\in A\}$ for some $a\in X$.

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