Proof of Frankl's conjecture on cross-intersecting families
Yongjiang Wu, Lihua Feng, Yongtao Li
TL;DR
The paper resolves Frankl's conjecture on the sum of sizes of cross-intersecting families under a restricted universe. It uses the shifting technique to reduce to shifted, restricted-universe families and an inductive framework to derive the tight bound $|\mathcal{F}|+|\mathcal{G}| \le \binom{k+t+s}{k+t} + \binom{n}{k} - \sum_{i=0}^s \binom{k+t+s}{i} \binom{n-k-t-s}{k-i}$. It also proves a parallel restricted-universe variant (Theorem main6) with a different projection parameter $m$, giving a bound $|\mathcal{F}|+|\mathcal{G}| \le \binom{n}{k} - \sum_{i=0}^s \binom{m}{i} \binom{n-m}{k-i}$ under the stated conditions, and provides corollaries for shifted and intersecting $\mathcal{G}$. Collectively, the results complete the conjecture and broaden the toolkit for extremal problems on cross-intersecting families with restricted universes.
Abstract
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 $\binom{[n]}{k}$ denote the family of all $k$-element subsets of $\{1,2,\ldots,n\}$. Let $t, s, k, n$ be non-negative integers with $k \geq s+1$ and $n \geq 2 k+t$. In 2016, Frankl proved that if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, and $\mathcal{F}$ is $(t+1)$-intersecting and $|\mathcal{F}| \geq 1$, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. Furthermore, Frankl conjectured that under an additional condition $\binom{[k+t+s]} {k+t}\subseteq\mathcal{F}$, the following inequality holds: $$ |\mathcal{F}|+|\mathcal{G}| \leq\binom{k+t+s}{k+t}+\binom{n}{k}-\sum_{i=0}^s\binom{k+t+s}{i}\binom{n-k-t-s}{k-i}. $$ 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.