The maximum product of sizes of cross-\(t\)-intersecting families
Jingjun Bao, Lijun Ji
TL;DR
This paper determines the maximum possible product of sizes for two cross-$t$-intersecting families of fixed uniformities, extending the Erdős–Ko–Rado framework. Using the shift operator and the generating-set method, it proves that for $3\le t\le l\le k$ and $\min\{m,n\}\ge (t+1)(k-t+1)$, the maximum product is $\binom{n-t}{k-t}\binom{m-t}{l-t}$, attained by $t$-stars around a common $t$-subset $T$. In the tight boundary case $n=m=(t+1)(k-t+1)$ with $k=l$, an additional extremal structure arises: $\mathcal{A}=\mathcal{F}(n,k,t,1)$ and $\mathcal{B}=\mathcal{F}(m,l,t,1)$ where $|A\cap T|\ge t+1$ for a fixed $(t+2)$-set $T$. The work confirms Tokushige's conjecture for $t\ge 3$ and provides a robust framework for cross-$t$-intersecting extremal problems via shifting, generating sets, and Sperner-type tools.
Abstract
Two families of sets \(\mathcal{A}\) and \(\mathcal{B}\) are called \emph{cross-\(t\)-intersecting} if \(|A \cap B| \geq t\) for all \(A \in \mathcal{A}\) and \(B \in \mathcal{B}\). Determining the maximum product of sizes for such cross-\(t\)-intersecting families is an active problem in extremal set theory. In this paper, we verify the following cross-\(t\)-intersecting version of the Erdős-Ko-Rado theorem: For \(k\geq l \geq t \geq 3\) and \(\min\{m,n\} \geq (t+1)(k-t+1)\), the maximun value of \(|\mathcal{A}||\mathcal{B}|\) for two cross-\(t\)-intersecting families \(\mathcal{A}\subseteq \binom{[n]}{k}\) and \(\mathcal{B} \subseteq \binom{[m]}{l}\) is \( \binom{n-t}{k-t}\binom{m-t}{l-t}\). Moreover, we characterize the extremal families attaining the upper bound. Our result confirms a conjecture of Tokushige for \(t \geq 3\), and actually proves a more general result.