发表机构
Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University（数学复杂系统与教育部重点实验室，北京师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
开发新方法处理极值集论交叉相交问题，基于迭代程序和t - 覆盖法，为交叉t - 相交族推导极值和稳定性结果，应用于多种组合对象，证明了相关定理。
AI 中文摘要
我们为极值集论中的交叉相交问题开发了一种新方法。该方法基于Kupavskii和Zakharov（2024）引入的迭代程序和t - 覆盖法。它为推导交叉t - 相交族的极值和稳定性结果提供了一个灵活的框架。我们的方法适用于各种组合对象。作为应用，我们为充分分散的集系统证明了开创性的埃尔德什 - 柯 - 拉多定理的乘积版本。
英文摘要
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 Erdős--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 Mörs (1985) and Füredi (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 $γ_t(\mathcal{F})$, 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 $γ_t$, and determine the maximum of $\min\{γ_t(\mathcal{F}),γ_t(\mathcal{G})\}$ 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).
