On nontrivial cross-2-intersecting families

Two families \(\mathcal{A}\subseteq\binom{[n]}{k}\) and \(\mathcal{B}\subseteq\binom{[n]}{\ell}\) are said to be nontrivial cross-\(t\)-intersecting if \(|A \cap B| \geq t\) for all \(A \in \mathcal{A}\) and \(B \in \mathcal{B}\), and $|\bigcap_{A\in \mathcal{A}\cup \mathcal{B}}A|<t$.

In this paper, we determine the upper bound on \(|\mathcal{A}||\mathcal{B}|\) of two nontrivial cross-\(2\)-intersecting families \(\mathcal{A}\subseteq\binom{[n]}{k}\) and \(\mathcal{B}\subseteq\binom{[n]}{\ell}\) for any positive integers $n,k,\ell$ with \(k\geq \ell \geq 3\) and \(n \geq 3(k-1)\).

Moreover, we characterize the extremal families attaining this bound.

This settles the last unsolved case of a recent result by He, Li, Wu and Zhang (J.

Combin.

Theory Ser.

A, 217 (2026) 106095).