On cross-2-intersecting families

Abstract

Two families $\mathcal{A}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ and $\mathcal{B}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{\ell }\right)$ are called cross-$t$-intersecting if $|A\cap B|\ge t$ for all $A\in \mathcal{A}$, $B\in \mathcal{B}$. Let $n$, $k$ and $\ell$ be positive integers such that $n\ge 3.38\ell$ and $\ell \ge k\ge 2$. In this paper, we will determine the upper bound of $|\mathcal{A}||\mathcal{B}|$ for cross-$2$-intersecting families $\mathcal{A}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ and $\mathcal{B}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{\ell }\right)$. The structures of the extremal families attaining the upper bound are also characterized. The similar result obtained by Tokushige can be considered as a special case of ours when $k=\ell$, but under a more strong condition $n>3.42k$. Moreover, combined with the results obtained in this paper, the complicated extremal structures attaining the upper bound for nontrivial cases can be relatively easy to reach with similar techniques.

This manuscripts was submitted to DAM on February 7, 2025