Intersections and Distinct Intersections in Cross-intersecting Families

Let $\mathcal{F},\mathcal{G}$ be two cross-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$. Let $\mathcal{F}\wedge \mathcal{G}$, $\mathcal{I}(\mathcal{F},\mathcal{G})$ denote the families of all intersections $F\cap G$ with $F\in \mathcal{F},G\in \mathcal{G}$, and all distinct intersections $F\cap G$ with $F\neq G, F\in \mathcal{F},G\in \mathcal{G}$, respectively. For a fixed $T\subset \{1,2,\ldots,n\}$, let $\mathcal{S}_T$ be the family of all $k$-subsets of $\{1,2,\ldots,n\}$ containing $T$. In the present paper, we show that $|\mathcal{F}\wedge \mathcal{G}|$ is maximized when $\mathcal{F}=\mathcal{G}=\mathcal{S}_{\{1\}}$ for $n\geq 2k^2+8k$, while surprisingly $|\mathcal{I}(\mathcal{F}, \mathcal{G})|$ is maximized when $\mathcal{F}=\mathcal{S}_{\{1,2\}}\cup \mathcal{S}_{\{3,4\}}\cup \mathcal{S}_{\{1,4,5\}}\cup \mathcal{S}_{\{2,3,6\}}$ and $\mathcal{G}=\mathcal{S}_{\{1,3\}}\cup \mathcal{S}_{\{2,4\}}\cup \mathcal{S}_{\{1,4,6\}}\cup \mathcal{S}_{\{2,3,5\}}$ for $n\geq 100k^2$. The maximum number of distinct intersections in a $t$-intersecting family is determined for $n\geq 3(t+2)^3k^2$ as well.