Non-empty pairwise cross-intersecting families

Abstract

Two families $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\ne \emptyset$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. We call $t$ families $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ pairwise cross-intersecting families if $\mathcal{A}_i$ and $\mathcal{A}_j$ are cross-intersecting when $1\le i<j \le t$. Additionally, if $\mathcal{A}_j\ne \emptyset$ for each $j\in [t]$, then we say that $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ are non-empty pairwise cross-intersecting. Let $\mathcal{A}_1\subset{[n]\choose k_1}, \mathcal{A}_2\subset{[n]\choose k_2}, \dots, \mathcal{A}_t\subset{[n]\choose k_t}$ be non-empty pairwise cross-intersecting families with $t\geq 2$, $k_1\geq k_2\geq \cdots \geq k_t$, $n\ge k_1+k_2$ and $d_1, d_2, \dots, d_t$ be positive numbers. In this paper, we give a sharp upper bound of $\sum_{j=1}^td_j|\mathcal{A}_j|$ and characterize the families $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ attaining the upper bound. Our results unifies results of Frankl and Tokushige [J. Combin. Theory Ser. A 61 (1992)], Shi, Frankl and Qian [Combinatorica 42 (2022)], Huang and Peng \cite{huangpeng}, and Zhang-Feng \cite{ZF2023}. Furthermore, our result can be applied in the treatment for some $n<k_1+k_2$ while all previous known results do not have such an application. In the proof, a result of Kruskal-Katona is applied to allow us to consider only families $\mathcal{A}_i$ whose elements are the first $|\mathcal{A}_i|$ elements in lexicographic order. We bound $\sum_{i=1}^t{|\mathcal{A}_i|}$ by a single variable function $g(R)$, where $R$ is the last element of $\mathcal{A}_1$ in lexicographic order. One crucial and challenge part is to verify that $-g(R)$ has unimodality. We think that the unimodality of functions in this paper are interesting in their own, in addition to the extremal result.