均匀超图的尖锐谱型Erdős–Ko–Rado定理
A Sharp Spectral Erdős--Ko--Rado Theorem for Uniform Hypergraphs
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中文总结 AI 辅助
该研究针对均匀超图的谱型Erdős–Ko–Rado问题，证明了t-相交k-均匀超图族的极值判定条件，给出了最优n范围及t=1时的所有极值结构，改进了相关定理的适用范围。
中文摘要 AI 辅助
谱型Erdős–Ko–Rado问题旨在求t-相交k-均匀超图族的邻接张量谱半径最大值。Keevash、Lenz与Mubayi证明，对固定的k、t及足够大的n，唯一的极值族是完整t-星，并询问此类定理是否可推广至所有n。设Frankl族为$\boldsymbol{\textit{A}}_r=\boldsymbol{\textit{F}}\boldsymbol{\textit{r}}\boldsymbol{\textit{a}}\boldsymbol{\textit{n}}\boldsymbol{\textit{k}}\boldsymbol{\textit{l}}\boldsymbol{\textit{ 族}}=\boldsymbol{\textit{F}}\boldsymbol{\textit{∈}}\boldsymbol{\textit{C}}\boldsymbol{\textit{([n],k)}}:\boldsymbol{\textit{|F∩[t+2r]|≥t+r}}$，其谱半径记为$\boldsymbol{\rho}_r$。对于$\boldsymbol{2≤t<k}$且$\boldsymbol{n>2k-t}$，我们证明$\boldsymbol{\textit{A}}_0$为谱型极值族当且仅当$\boldsymbol{\rho}_0≥\boldsymbol{\rho}_1}$；若不等式严格成立，则除置换外唯一，相等时$\boldsymbol{\textit{A}}_0$与$\boldsymbol{\textit{A}}_1$均为极值族。Ahlswede–Khachatrian基数证明中使用的分层收缩法不适用于此处，直接应用可能降低谱半径，我们转而同时收缩所有边界层并应用Perron尾对称化。由此可得，当$\boldsymbol{n≥(t+1)(k-t+1)+⌈(t+1)log(t+1)⌉}$时，$\boldsymbol{\textit{A}}_0$为唯一极值族；对固定t，主导系数$\boldsymbol{t+1}$是最优的。我们还确定了$\boldsymbol{t=1}$且$\boldsymbol{n≥2k}$范围内的所有极值结构。
英文摘要
The spectral Erdős--Ko--Rado problem asks for the largest adjacency-tensor spectral radius of a $t$-intersecting $k$-uniform family. Keevash, Lenz and Mubayi proved that, for fixed $k,t$ and sufficiently large $n$, the unique extremal family is a full $t$-star, and asked whether such a theorem extends to all $n$. Let $\mathcal{A}_r=\{F\in\binom{[n]}k:|F\cap[t+2r]|\ge t+r\}$ be the Frankl families and write $ρ_r$ for their spectral radii. For $2\le t<k$ and $n>2k-t$, we prove that $\mathcal{A}_0$ is spectrally extremal if and only if $ρ_0\geρ_1$; it is unique up to permutation when the inequality is strict, whereas $\mathcal{A}_0$ and $\mathcal{A}_1$ are both extremal at equality. The layerwise pull used in the Ahlswede--Khachatrian cardinality proof is not applicable here: applied directly, it may decrease the spectral radius. Our proof instead pulls all boundary layers simultaneously and applies Perron tail symmetrization. It follows that $\mathcal{A}_0$ is uniquely extremal for $n\ge (t+1)(k-t+1)+\lceil(t+1)\log(t+1)\rceil$; the leading coefficient $t+1$ is best possible for fixed $t$. We also determine all extremal structures for $t=1$ throughout the range $n\ge2k$.
