关于希尔顿 - 米尔纳定理乘积版本的一个猜想
On a conjecture regarding the product version of the Hilton-Milner theorem
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中文总结 AI 辅助
研究Frankl和Wang关于希尔顿 - 米尔纳定理乘积版本的猜想，通过双中心构造证其在特定线性范围不成立，又证明在$n > 100\ell k^2$且$3 \leq \ell < k$时成立并刻画极值族，依赖最小覆盖大小和2 - 覆盖图结构性质。
中文摘要 AI 辅助
最近，Frankl和Wang考虑了经典希尔顿 - 米尔纳定理的乘积版本。他们猜想，如果$\mathcal{F} \subset \binom{[n]}{k}$且$\mathcal{G} \subset \binom{[n]}{\ell}$是非平凡交叉相交族，其中$n \geq 2k > 2\ell \geq 4$，那么$|\mathcal{F}||\mathcal{G}|$的最大值由自然的希尔顿 - 米尔纳型配置达到。本文给出了关于这个猜想的两个主要结果。首先，通过引入双中心构造证明对于每个固定整数$\ell \geq 3$和所有足够大的$k$，在$2k + 1 \leq n \leq (c_\ell - \epsilon)k$（对于任意$0 < \epsilon < c_\ell - 2$，其中$c_\ell > 2$是明确常数）的线性范围内猜想不成立。其次，证明当$n > 100\ell k^2$且$3 \leq \ell < k$时猜想成立，并完全刻画了极值族。证明依赖于最小覆盖的大小和分析2 - 覆盖图的结构性质。
英文摘要
Recently, Frankl and Wang considered a product version of the classical Hilton-Milner theorem. They conjectured that, if $\mathcal{F} \subset \binom{[n]}{k}$ and $\mathcal{G} \subset \binom{[n]}{\ell}$ are non-trivial cross-intersecting families with $n \geq 2k > 2\ell \geq 4$, 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 $\ell \geq 3$ and all sufficiently large $k$, the conjecture is false in a linear range $2k+1 \leq n \leq (c_\ell - ε)k$ for any $0 < ε< c_\ell - 2$, where $c_\ell > 2$ is an explicit constant. Secondly, we prove that the conjecture holds when $n > 100\ell k^2$ and $3 \leq \ell < k$, 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.
