Non-empty pairwise cross-intersecting families

Abstract

Two families $A$ and $B$ are cross-intersecting if $A\cap B\ne \varnothing$ for any $A\in A$ and $B\in B$. We call t families $A_1,A_2,\dots ,A_t$ pairwise cross-intersecting families if $A_i$ and $A_j$ are cross-intersecting for $1\le i<j\le t$. Additionally, if $A_j\ne \varnothing$ for each $j\in \left[t\right]$, then we say that $A_1,A_2,\dots ,A_t$ are non-empty pairwise cross-intersecting. Let $A_1\subseteq \left(\begin{array}{c}\left[n\right]\\ k_1\end{array}\right),A_2\subseteq \left(\begin{array}{c}\left[n\right]\\ k_2\end{array}\right),\dots ,A_t\subseteq \left(\begin{array}{c}\left[n\right]\\ k_t\end{array}\right)$ be non-empty pairwise cross-intersecting families with $t\ge 2$, $k_1\ge k_2\ge \cdots \ge k_t$, $n\ge k_1+k_2$ and $d_1,d_2,\dots ,d_t$ be positive numbers. In this paper, we give a sharp upper bound of ${\sum }_{j=1}^t{d}_j\left|A_j\right|$ and characterize the families $A_1,A_2,\dots ,A_t$ attaining the upper bound. Our results unifies results of Frankl and Tokushige (1992) [5], Shi, Frankl and Qian (2022) [15], Huang, Peng and Wang [10], and Zhang and Feng [16]. 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 and Katona is applied to allow us to consider only families $A_i$ whose elements are the first $\left|A_i\right|$ elements in lexicographic order. We bound ${\sum }_{i=1}^t{d}_i\left|A_i\right|$ by a single variable function $f_i\left(R\right)$, where R is the last element of $A_i$ in lexicographic order, and verify that $-f_i\left(R\right)$ has unimodality which is stronger than the extremal result. We think that the unimodality of functions in this paper is interesting in its own, in addition to the extremal result.