Non-uniform Cross-intersecting Families (2411.18426v1)

Abstract: 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.