大型t-相交族的结构 I：Hilton–Milner–Frankl定理的稳定性

中文摘要 AI 辅助

我们研究大型t-相交族的结构。n元集合的k-子集族是t-相交的，当且仅当其任意两个成员的交集至少包含t个元素。若一个t-相交族不包含任何一个t-子集属于其所有成员，则称其为非平凡的。我们对开创性的Hilton–Milner–Frankl定理证明了若干稳定性结果。首先，对任意固定的η、ε、θ∈(0,1)，我们证明：当k/t≥1+η且n=Ω(tk^(1+ε))时，每个大小大于(1+θ)|𝒦|的非平凡t-相交族，都是该定理中两个极端族之一的子族，其中𝒦是一个明确的大型非平凡t-相交族。证明中的关键要素是一个删除引理。我们还获得了所有大小不小于𝒦减去一个明确低阶项的t-相交族的分类，前提是k≥t+4≥6且n≥t+6·max{(t+2)², k(k−t)}。这在广泛的k和t范围内（例如当k−t≥2√t时），改进了Cao–Lv–Wang（2021）和Frankl（2025）的结果。作为该分类的一个应用，我们针对每个不超过t(n−k)的t-多样性预设下界，确定了最大的t-相交族，从而得到了Han和Kohayakawa（2017）以及Kupavskii（2025）结果的t-相交版本。为建立这些结果，我们开发了基于扩散近似方法和t-覆盖方法的技术，这些技术可能对其他相交问题有用。

英文摘要

We study the structure of large $t$-intersecting families. A family of $k$-subsets of an $n$-set is $t$-intersecting if every two of its members intersect in at least $t$ elements. A $t$-intersecting family is non-trivial if no $t$-subset is contained in all its members. We prove several stability results for the seminal Hilton--Milner--Frankl theorem. First, for any fixed $η,\varepsilon,θ\in(0,1)$, we prove that if $k/t\geq1+η$ and $n=Ω(tk^{1+\varepsilon})$, then every non-trivial $t$-intersecting family of size greater than $(1+θ)|\mathcal{K}|$ is a subfamily of one of the two extremal families in the theorem, where $\mathcal{K}$ is an explicit large non-trivial $t$-intersecting family. The key ingredient in the proof is a removal lemma. We also obtain a classification of all $t$-intersecting families with size bounded below by $|\mathcal{K}|$ minus an explicit lower-order term, provided that $k\geq t+4\geq6$ and $n\geq t+6\cdot\max\{(t+2)^2, k(k-t)\}$. This strengthens results of Cao--Lv--Wang (2021) and Frankl (2025) for a broad range of $k$ and $t$ (for example, when $k-t\geq2\sqrt{t}$). As an application of this classification, we determine the largest $t$-intersecting families for each prescribed lower bound on $t$-diversity not exceeding $t(n-k)$, thereby obtaining $t$-intersection versions of results of Han and Kohayakawa (2017) and Kupavskii (2025). To establish these results, we develop techniques based on the spread approximation method and the $t$-cover method, which may be useful for other intersection problems.

发表机构

• School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）
• Laboratory of Mathematics and Complex Systems (Ministry of Education)（数学复杂系统与先进测量教育部重点实验室）

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