A sharp product bound for non-trivial cross-intersecting families

Abstract: Two families $\mathcal{A}, \mathcal{B} \subset \binom{[n]}{k}$ are cross-intersecting if $A \cap B \ne \emptyset$ for all $A \in \mathcal{A}$ and $B \in \mathcal{B}$, and non-trivial if neither $\m A$ nor $\m B$ is a star. Pyber proved that any two cross-intersecting families $\mathcal{A}, \mathcal{B} \subset \binom{[n]}{k}$ satisfy $|\mathcal{A}||\mathcal{B}| \le \binom{n-1}{k-1}^2$, and the maximum is attained by two full stars. Frankl, as well as Frankl and Wang, conjectured that the sharp bound, when both families are required to be non-trivial, is $h(n,k)^2$, where $h(n,k) = \binom{n-1}{k-1} - \binom{n-k-1}{k-1} + 1$, the size of the Hilton--Milner family. The cases $k=3$, and the range $k\ge8$ and $n\ge 4k$, were established earlier by Frankl and by Frankl and Wang, respectively. In this paper, we prove their conjecture in the full range. We show that every non-trivial cross-intersecting pair $\mathcal{A}, \mathcal{B} \subset \binom{[n]}{k}$ with $n \ge 2k$ and $k \ge 3$ satisfies $|\mathcal{A}||\mathcal{B}| \le h(n,k)^2$. Moreover, we characterize all extremal pairs. Whereas the corresponding sum problem admits asymmetric and unbalanced extremizers, the product extremum forces a balanced, symmetric-or-dual structure: the two families are isomorphic when $n>2k$ and complement-dual when $n=2k$. Independently and contemporaneously with the present work, Frankl and Wang obtained the same bound for $k\ge8$ and $n\ge2k+1$ by a different method. Our proof combines a diversity technique with several new properties of an extended shift operation. Moreover, we show that the problem behaves differently for different uniformities, exhibiting new extremal configurations. In particular, we disprove a related conjecture proposed by Frankl and Wang.