Complete Characterization on Maximum Pairwise Cross Intersecting Families (I)
Yang Huang, Yuejian Peng
TL;DR
This work addresses the problem of determining $M(n, k_1, \dots, k_t)$, the maximum total size of non-empty pairwise cross-intersecting families with $\mathcal{F}_i \subseteq {[n] \choose k_i}$, in the mixed-type regime $k_1+k_3 \le n < k_1+k_2$. The authors develop a lexicographic (L-initial) reduction via the Kruskal–Katona theorem and introduce the notions of $k$-partner, parity, and corresponding $k$-set to bound the total size by a single-variable function $f(I_1)$ determined by the ID $I_1$ of $\mathcal{F}_1$, with $I_2$ forced to be the corresponding $k_2$-set of $I_1$. They prove that extremal configurations are isomorphic to two canonical constructions (Examples con1 and con2), except for a specific exceptional case, thereby completing the mixed-type characterization in this range. The paper also establishes foundational lemmas (notably Lemma claim4) that facilitate a broader program toward the general constraint $n \ge k_1+k_t$, and provides a framework potentially applicable to future generalizations in extremal set theory. This advances the understanding of cross-intersecting multi-families beyond the previously resolved non-mixed type, with implications for related combinatorial optimization problems.
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 $n\geq k_1+k_2$.