Proof of Frankl's conjecture on cross-intersecting families

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. For any positive integers $n$ and $k$, let $\left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ denote the family of all $k$-element subsets of $\{1,2,\dots ,n\}$. Let $t,s,k,n$ be non-negative integers with $k\ge s+1$ and $n\ge 2k+t$. In 2016, Frankl proved that if $\mathcal{F}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k+t}\right)$ and $\mathcal{G}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ are cross-intersecting families, and $\mathcal{F}$ is $(t+1)$-intersecting and $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\ge 1$, then $\mathrm{\mid }\mathcal{F}\mathrm{\mid }+\mathrm{\mid }\mathcal{G}\mathrm{\mid }\le \left(\genfrac{}{}{0px}{}{n}{k}\right)-\left(\genfrac{}{}{0px}{}{n-k-t}{k}\right)+1$. Furthermore, Frankl conjectured that under an additional condition $\left(\genfrac{}{}{0px}{}{[k+t+s]}{k+t}\right)\subseteq \mathcal{F}$, the following inequality holds: $$\mathrm{\mid }\mathcal{F}\mathrm{\mid }+\mathrm{\mid }\mathcal{G}\mathrm{\mid }\le \left(\genfrac{}{}{0px}{}{k+t+s}{k+t}\right)+\left(\genfrac{}{}{0px}{}{n}{k}\right)-\sum _{i=0}^s\left(\genfrac{}{}{0px}{}{k+t+s}{i}\right)\left(\genfrac{}{}{0px}{}{n-k-t-s}{k-i}\right)\mathrm{.}$$ In this paper, we prove this conjecture. The key ingredient is to establish a theorem for cross-intersecting families with a restricted universe. Moreover, we derive an analogous result for this conjecture.