Intersecting families in a subset of Boolean lattices (Q665759)

Summary: 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, both of size \(n\). Define \[ \mathcal{F} = \{A \in \binom{X}{r+\ell}: |A \cap X_{1}| = r \text{ or }\ell\}, \] 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 \( a \in X\).