On nontrivial cross-2-intersecting families
arXiv:2607.12239
The paper determines the maximum possible product of the sizes of two nontrivial cross‑2‑intersecting families of k‑ and ℓ‑subsets of an n‑element set, and describes the families that achieve this bound.
Abstract
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 . 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 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).