Complete Characterization on Maximum Pairwise Cross Intersecting Families (I)

TL;DR

This paper fully characterizes the maximum total size of multiple cross-intersecting families of subsets with given sizes, extending previous results and solving an open problem for all sufficiently large n.

Contribution

It provides a complete characterization of maximum sum sizes for multi-family cross-intersecting systems, generalizing prior two-family results and resolving an open problem.

Findings

Determined $M(n,k_1, imes, k_t)$ for all $n extgreater k_1 + k_2$.

Extended the understanding of cross-intersecting families to multiple families.

Unified previous partial results into a comprehensive characterization.

Abstract

The families $\mathcal{A}$ and $\mathcal{B}$ are cross intersecting if $A\cap B\ne \emptyset$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. Let $t\geq 2$ and $k_1\geq k_2\geq \cdots \geq k_t$. We say that $(\mathcal{F}_1, \dots, \mathcal{F}_t)$ is an $(n, k_1, \dots, k_t)$-cross intersecting system if $\mathcal{F}_1 \subseteq{[n]\choose k_1}, \ldots ,\mathcal{F}_t \subseteq{[n]\choose k_t}$ are non-empty pairwise cross intersecting families. Let $M(n,k_1,\ldots ,k_t)$ denote the maximum sum of sizes of families of an $(n,k_1,\ldots ,k_t)$-cross intersecting system. The case $t=2$ was studied by Frankl--Tokushige. Solving a problem of Shi-Frankl-Qian, Huang-Peng-Wang and Zhang-Feng independently determined $M(n, k_1, \dots, k_t)$ for all $n\geq k_1+k_2$.