Proof of Frankl's conjecture on cross-intersecting families

Abstract

Two families $F$ and $G$ are called cross-intersecting if for every $F\in F$ and $G\in G$, the intersection $F\cap G$ is non-empty. For any positive integers n and k, let $\left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$ 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 $F\subseteq \left(\begin{array}{c}\left[n\right]\\ k+t\end{array}\right)$ and $G\subseteq \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$ are cross-intersecting families, and $F$ is $(t+1)$-intersecting and $\left|F\right|\ge 1$, then $\left|F\right|+\left|G\right|\le \left(\begin{array}{c}n\\ k\end{array}\right)-\left(\begin{array}{c}n-k-t\\ k\end{array}\right)+1$. Furthermore, Frankl conjectured that under an additional condition $\left(\begin{array}{c}[k+t+s]\\ k+t\end{array}\right)\subseteq F$, the following inequality holds:$\left|F\right|+\left|G\right|\le \left(\begin{array}{c}k+t+s\\ k+t\end{array}\right)+\left(\begin{array}{c}n\\ k\end{array}\right)-\sum _{i=0}^s\left(\begin{array}{c}k+t+s\\ i\end{array}\right)\left(\begin{array}{c}n-k-t-s\\ k-i\end{array}\right).$ 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.