A unified approach to cross-intersection problems with applications to Hilton--Milner type theorems and stability

Abstract

We develop a new approach to cross-intersection problems in extremal set theory. The method builds on the iterative procedure introduced by Kupavskii and Zakharov (2024) and the $t$-cover method. It provides a flexible framework for deriving extremal and stability results for cross $t$-intersecting families. Our approach applies to a variety of combinatorial objects. As an application, we prove a product version of the seminal Erd\H{o}s--Ko--Rado theorem for sufficiently spread set systems. Two families $\mathcal{F}$ and $\mathcal{G}$ of $k$-subsets of $[n]$ are called cross $t$-intersecting if $|F\cap G|\geq t$ for all $F\in\mathcal{F}$ and $G\in\mathcal{G}$. We determine the families maximizing $\min\{|\mathcal{F}|, |\mathcal{G}|\}$ for large $n$ and all $t\ge2$, generalizing results of M\"{o}rs (1985) and F\"{u}redi (1995) for cross $1$-intersecting families. We then determine the families maximizing $|\mathcal{F}||\mathcal{G}|$ under the condition $\max\{|\cap_{F\in\mathcal{F}}F|,|\cap_{G\in\mathcal{G}}G|\}<t$ for large $n$. This improves the bound obtained by Frankl and Wang (2024), and provides a characterization of extremal configurations. For a family $\mathcal{F}$ of subsets of $[n]$, we introduce its $t$-diversity $\gamma_t(\mathcal{F})$, defined as the minimum number of sets from $\mathcal{F}$ not containing a fixed $t$-subset. This serves as a natural generalization of the important notion of diversity for $t=1$. We obtain a stability result via $\gamma_t$, and determine the maximum of $\min\{\gamma_t(\mathcal{F}),\gamma_t(\mathcal{G})\}$ for cross $t$-intersecting families $\mathcal{F}$ and $\mathcal{G}$. These yield new results for $t$-intersecting families, including a stability theorem towards a conjecture of Ellis, Keller and Lifshitz (2019), which may also be regarded as a $t$-intersection version, for large $n$, of an influential theorem of Frankl (1987).