Complete Characterization on Maximum Pairwise Cross Intersecting Families (I)

The families $\mathcal{A}$ and $\mathcal{B}$ are cross intersecting if $A\cap B\mathrm{\ne }\mathrm{\varnothing }$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. Let $t\ge 2$ and $k_1\ge k_2\ge \cdots \ge k_t$. We say that $({\mathcal{F}}_1,\dots ,{\mathcal{F}}_t)$ is an $(n,k_1,\dots ,k_t)$-cross intersecting system if ${\mathcal{F}}_1\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k_1}\right),\dots ,{\mathcal{F}}_t\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k_t}\right)$ are non-empty pairwise cross intersecting families. Let $M(n,k_1,\dots ,k_t)$ denote the maximum sum of sizes of families of an $(n,k_1,\dots ,k_t)$-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 $n\ge k_1+k_2$.