On a conjecture regarding the product version of the Hilton-Milner theorem

Abstract

Recently, Frankl and Wang considered a product version of the classical Hilton-Milner theorem. They conjectured that, if $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ and $\mathcal{G}\subset \left(\genfrac{}{}{0ex}{}{[n]}{\ell }\right)$ are non-trivial cross-intersecting families with $n\ge 2k>2\ell \ge 4$, the maximum of $|\mathcal{F}||\mathcal{G}|$ is attained by the natural Hilton-Milner-type configurations. In this paper, we present two main results concerning this conjecture. Firstly, we show that the conjecture does not hold in general. By introducing a two-center construction, we prove that for every fixed integer $\ell \ge 3$ and all sufficiently large $k$, the conjecture is false in a linear range $2k+1\le n\le (c_{\ell }-\epsilon )k$ for any $0<\epsilon <c_{\ell }-2$, where $c_{\ell }>2$ is an explicit constant. Secondly, we prove that the conjecture holds when $n>100\ell k^2$ and $3\le \ell <k$, and we completely characterize the extremal families. Our proofs rely on the size of minimal covers and analyzing the structural properties of $2$-cover graphs.