The maximum sum of the size of all intersections within intersecting families and crossing-intersecting families

Abstract

Let $\omega(\mathcal{F})=\sum_{\{A,B\}\subset\mathcal{F}}|A\cap B|$ and $\omega(\mathcal{A},\mathcal{B})=\sum_{(A,B)\in \mathcal{A}\times \mathcal{B}}|A\cap B|$. A family $\mathcal{F}$ is intersecting if $F_1\cap F_2\neq \emptyset$ for any $F_1,F_2\in\mathcal{F}$ and two family $\mathcal{A}$ and $\mathcal{B}$ are crossing-intersecting if $A\cap B\neq \emptyset$ for any $(A,B)\in \mathcal{A}\times\mathcal{B}$. For an intersecting family $\mathcal{F}$, Erd\H{o}s, Ko and Rado determined the upper bound of $|\mathcal{F}|$, consequently yielding an upper bound of $\binom{|\mathcal{F}|}{2}=\sum_{\{A,B\}\subset\mathcal{F}}1$. If we replace $1$ with $|A\cap B|$ in the summation $\sum_{\{A,B\}\subset\mathcal{F}}1$, then this summation transforms into $\omega(\mathcal{F})$. In this paper, for an intersecting family $\mathcal{F}$, we determine the upper bound of $\omega(\mathcal{F})$, which is a generalization of Erd\H{o}s-Ko-Rado Theorem. Further, for crossing-intersecting families $\mathcal{A}$ and $\mathcal{B}$, we determine the upper bound of $\omega(\mathcal{A},\mathcal{B})$.