Non-uniform pairwise cross $t$-intersecting families

Abstract

Let $n\ge t\ge 1$ and ${\mathcal{A}}_1,{\mathcal{A}}_2,\dots ,{\mathcal{A}}_m\subseteq 2^{[n]}$ be non-empty families. We say that they are pairwise cross $t$-intersecting if $|A_i\cap A_j|\ge t$ holds for any $A_i\in {\mathcal{A}}_i$ and $A_j\in {\mathcal{A}}_j$ with $i\ne j$. In the case where $m=2$ and ${\mathcal{A}}_1={\mathcal{A}}_2$, determining the maximum size $M(n,t)$ of a non-uniform $t$-intersecting family of sets over $[n]$ was solved by Katona (1964), and enhanced by Frankl (2017), and recently by Li and Wu (2024). In this paper, we establish the following upper bound: if ${\mathcal{A}}_1,{\mathcal{A}}_2,\dots ,{\mathcal{A}}_m\subseteq 2^{[n]}$ are non-empty pairwise cross $t$-intersecting families, then $$\sum _{i=1}^m|{\mathcal{A}}_i|\le \max \left\{\sum _{k=t}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)+m-1,mM(n,t)\right\}.$$ Furthermore, we provide a complete characterization of the extremal families that achieve the bound. Our result not only generalizes an old result of Katona (1964) for a single family, but also extends a theorem of Frankl and Wong (2021) for two families. Moreover, our result could be viewed as a non-uniform version of a recent theorem of Li and Zhang (2025). The key in our proof is to utilize the generating set method and the pushing-pulling method together.

17 pages, We corrected some mistakes