On a conjecture of Tokushige for cross-$t$-intersecting families

TL;DR

This paper proves a conjecture in extremal set theory, establishing the maximum product size of cross-$t$-intersecting families of sets under certain conditions, extending the classical Erdős–Ko–Rado problem.

Contribution

It confirms Tokushige's conjecture for $t ext{-}ge 3$, providing a tight bound on the product of sizes of cross-$t$-intersecting families.

Findings

Maximum product of sizes is ${inom{n-t}{k-t}}^2$ under specified conditions.

Equality holds if and only if the families are equal and maximum $t$-intersecting.

Results extend classical intersection theorems to cross-$t$-intersecting families.

Abstract

Two families of sets $\mathcal{A}$ and $\mathcal{B}$ are called cross-$t$-intersecting if $|A\cap B|\ge t$ for all $A\in \mathcal{A}$, $B\in \mathcal{B}$. An active problem in extremal set theory is to determine the maximum product of sizes of cross-$t$-intersecting families. This incorporates the classical Erd\H{o}s--Ko--Rado (EKR) problem. In the present paper, we prove that if $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting families of $\binom {[n]}k$ with $k\ge t\ge 3$ and $n\ge (t+1)(k-t+1)$, then $|\mathcal{A}||\mathcal{B}|\le {\binom{n-t}{k-t}}^2$; moreover, if $n>(t+1)(k-t+1)$, then equality holds if and only if $\mathcal{A}=\mathcal{B}$ is a maximum $t$-intersecting subfamily of $\binom{[n]}{k}$. This confirms a conjecture of Tokushige for $t\ge 3$.