The maximum product of sizes of cross-$t$-intersecting families

Jingjun Bao; Lijun Ji

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arXiv · 2510.11724

The maximum product of sizes of cross-$t$-intersecting families

Jingjun Bao·Lijun Ji

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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 Erd\H{o}s-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.

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BibTeX RIS

Jingjun Bao, Lijun Ji. 2025-10-07. The maximum product of sizes of cross-$t$-intersecting families. https://arxiv.org/abs/2510.11724

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