Exact extremal non-trivial cross-intersecting families

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are called cross-intersecting if each pair of sets $A\in \mathcal{A}$ and $B\in \mathcal{B}$ has nonempty intersection. Let $\mathcal{A}$ and $\mathcal{B}$ be two cross-intersecting families of $k$-subsets and $\mathrm{\ell }$-subsets of $[n]$. Matsumoto and Tokushige [J. Combin. Theory Ser. A 52 (1989) 90–97] studied the extremal problem of the size $\mathrm{\mid }\mathcal{A}\mathcal{\mid }\mathcal{\mid }\mathcal{B}\mathcal{\mid }$ and obtained the uniqueness of extremal families whenever $n\ge 2\mathrm{\ell }\ge 2k$, building on the work of Pyber. This paper will explore the second extremal size of $\mathrm{\mid }\mathcal{A}\mathcal{\mid }\mathcal{\mid }\mathcal{B}\mathcal{\mid }$ and obtain that if $\mathcal{A}$ and $\mathcal{B}$ are not the subfamilies of Matsumoto–Tokushige’s extremal families, then, for $n\ge 2\mathrm{\ell }>2k$ or $n>2\mathrm{\ell }=2k$, \begin{itemize} \item[1)]either $\mathrm{\mid }\mathcal{A}\mathcal{\mid }\mathcal{\mid }\mathcal{B}\mathcal{\mid }\le (\left(\genfrac{}{}{0px}{}{\mathcal{n}-\mathcal{1}}{\mathcal{k}-\mathcal{1}}\right)+\left(\genfrac{}{}{0px}{}{\mathcal{n}-\mathcal{2}}{\mathcal{k}-\mathcal{1}}\right))\left(\genfrac{}{}{0px}{}{\mathcal{n}-\mathcal{2}}{\mathcal{\ell }-\mathcal{2}}\right)$ with the unique extremal families (up to isomorphism) $\mathcal{A=\{A\in {\binom{[n]}{k}}: 1\in A \: \mathrm{ or} \: 2\in A\}$ \quad and \quad $\mathcal{B}=\{B\in {\binom{[n]}{\ell}}: [2] \subseteq B\}$}; \item[2)] or $\mathrm{\mid }\mathcal{A}\mathcal{\mid }\mathcal{\mid }\mathcal{B}\mathcal{\mid }\le (\left(\genfrac{}{}{0px}{}{\mathcal{n}-\mathcal{1}}{\mathcal{k}-\mathcal{1}}\right)+\mathcal{1})(\left(\genfrac{}{}{0px}{}{\mathcal{n}-\mathcal{1}}{\mathcal{\ell }-\mathcal{1}}\right)-\left(\genfrac{}{}{0px}{}{\mathcal{n}-\mathcal{k}-\mathcal{1}}{\mathcal{\ell }-\mathcal{1}}\right))$ with the unique extremal families (up to isomorphism) $\mathcal{A=\{A\in {\binom{[n]}{k}}: 1\in A\}\cup \{[2,k+1] \}$\quad and \quad $\mathcal{B}=\{B\in {\binom{[n]}{\ell}}: 1\in B, B\cap [2,k+1]\neq \emptyset \}$.} \end{itemize} The bound ``$n\ge 2\mathrm{\ell }>2k$ or $n>2\mathrm{\ell }=2k$" is sharp for $n$. To achieve the above results, we establish some size-sensitive inequalities for cross-intersecting families. As by-products, we will recover the main results of Frankl and Kupavskii [European J. Combin. 62 (2017) 263–271].