On a conjecture of Tokushige for 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 Erdős--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 $t\ge 3$.

Theorems & Definitions (23)

• Theorem 1.1: Erdős--Ko--Rado Theorem EKR1961
• Theorem 1.2: Ahlswede and Khachatrian AK
• Theorem 1.3: Pyber Pyber
• Theorem 1.4: Matsumoto and Tokushige MT1989
• Theorem 1.5: Tokushige Tokushige2013
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 2.3: Wang and Zhangwz2