Non-empty pairwise cross-intersecting families

TL;DR

This paper establishes a sharp upper bound for the weighted sum of sizes of non-empty pairwise cross-intersecting families of sets, generalizing previous results and allowing applications for smaller set sizes than previously possible.

Contribution

It provides a unified extremal bound for such families, characterizes the extremal families, and introduces novel techniques involving lexicographic order and unimodality analysis.

Findings

Sharp upper bound for sum of weighted family sizes

Characterization of extremal families achieving the bound

Application to cases with smaller n than k_1 + k_2

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.