Let $m\geq 2$, $n$ be positive integers, and $R_i=\{ k_{ i,1} >k_{ i,2} >\cdots> k_{ i,t_i}\}$ be subsets of $[n]$ for $i=1,2,\ldots,m$. The families $\mathcal{ F}_1\subseteq \binom{ [n]}{ R_1},\mathcal{ F}_2\subseteq \binom{ [n]}{ R_2},\ldots,\mathcal{ F}_m\subseteq \binom{ [n]}{ R_m}$ are said to be non-empty cross-intersecting if for each $i\in [m]$, $\mathcal{ F}_i\neq\emptyset$ and for any $A\in \mathcal{ F}_i,B\in\mathcal{ F}_j$, $1\leq i<j\leq m$, $|A\bigcap B|\geq1$. In this paper, we determine the maximum value of $\sum_{ j=1}^{ m}|\mathcal{ F}_j|$ for non-empty cross-intersecting family $\mathcal{ F}_1, \mathcal{ F}_2,\ldots,\mathcal{ F}_m$ when $n\geq k_1+k_2$, where $k_1$ (respectively, $k_2$) is the largest (respectively, second largest) value in $\{ k_{ 1,1},k_{ 2,1},\ldots,k_{ m,1}\}$. This result is a generalization of the results by Shi, Frankl and Qian \cite{ shi2022non} on non-empty cross-intersecting families. Moreover, the extremal families are completely characterized.
Categories: math.CO