AI 中文总结
本文研究交叉相交族的Erdős–Ko–Rado定理，先结合Filmus FKN定理与线性代数方法给出经典结果的稳定性结论，再基于此与Johnson方案特征值，提出欧几里得内积下的对应定理及唯一性、稳定性结果。
AI 中文摘要
设$\binom{[n]}{k}$为集合$\{1,\ldots,n\}$的所有$k$元子集构成的集合，$\mathcal A,\mathcal B \subseteq \binom{[n]}{k}$为两个交叉相交族，即对任意$A\in \mathcal A$和$B\in \mathcal B$，都有$A\cap B\neq \emptyset$。由Pyber与Matsumoto–Tokushige提出的经典Erdős–Ko–Rado定理交叉相交版本指出，若$n\geq 2k$，则$|\mathcal A||\mathcal B|\leq \binom{n-1}{k-1}^2$；当$n>2k$时，等号成立当且仅当$\mathcal A=\mathcal B$是一个星族。本文首先利用切片上的Filmus FKN定理与线性代数方法，给出该定理的一个稳定性结果：存在常数$C>1$，使得若$n\geq 2.07k$且$|\mathcal A||\mathcal B|\geq (1-ε)\binom{n-1}{k-1}^2$（其中$ε\leq \frac{k^2}{C^2 n ^2 }$），则存在星族$\mathcal{S}$，满足$|\mathcal{S} Δ\mathcal A|\leq C ε\binom{n}{k}$且$|\mathcal{S} Δ\mathcal B|\leq C ε\binom{n}{k}$。此外，基于该稳定性结果与Johnson方案的矩阵特征值，我们提出了欧几里得内积下交叉相交族的Erdős–Ko–Rado定理，证明若$n\geq 2k$且$k\geq d \geq 0$，则$$\big\langle\mathbf{v}_d(\mathcal A),\mathbf{v}_d(\mathcal B)\big\rangle \leq \frac{\binom{k}{d}\binom{k-1}{d}}{\binom{n-1}{d}}\binom{n-1}{k-1}^2 +\binom{k-1}{d-1} \binom{n-d-1}{k-d}\binom{n-1}{k-1},$$同时给出了唯一性结论与对应的稳定性结果。其中$\mathbf{v}_d(\mathcal A) \in \mathbb R^{\binom{[n]}{d}}$是$\mathcal A$的$d$次向量，其$U$位置的元素为$\mathcal A$中包含$U$的成员数量。
英文摘要
Let $\binom{[n]}{k}$ be the set of all $k$-element subsets of the set $\{1,\ldots,n\}$ and let $\mathcal A,\mathcal B \subseteq \binom{[n]}{k}$ be two cross-intersecting families, that is, $A\cap B\neq \emptyset$ for any $A\in \mathcal A$ and $B\in \mathcal B$. The classical cross-intersecting version of the Erdős--Ko--Rado theorem, due to Pyber and Matsumoto--Tokushige, states that if $n\geq 2k$, then $|\mathcal A||\mathcal B|\leq \binom{n-1}{k-1}^2,$ where the equality holds for $n>2k$ if and only if $\mathcal A=\mathcal B$ is a star. In the present paper, we first give a stability result of this theorem by using Filmus's FKN theorem on the slice and linear algebra method as follows: There exists a constant $C>1$ such that if $n\geq 2.07k$ and $|\mathcal A||\mathcal B|\geq (1-ε)\binom{n-1}{k-1}^2$, where $ε\leq \frac{k^2}{C^2 n ^2 }$, then there is a star $\mathcal{S}$ such that $|\mathcal{S} Δ\mathcal A|\leq C ε\binom{n}{k}$ and $|\mathcal{S} Δ\mathcal B|\leq C ε\binom{n}{k}.$ Moreover, based on this stability result and the eigenvalues of the matrices of the Johnson scheme, we present an Erdős--Ko--Rado theorem for cross-intersecting families in the Euclidean inner product showing that if $n\geq 2k$ and $k\geq d \geq 0$, then $$\big\langle\mathbf{v}_d(\mathcal A),\mathbf{v}_d(\mathcal B)\big\rangle \leq \frac{\binom{k}{d}\binom{k-1}{d}}{\binom{n-1}{d}}\binom{n-1}{k-1}^2 +\binom{k-1}{d-1} \binom{n-d-1}{k-d}\binom{n-1}{k-1},$$ together with uniqueness and a corresponding stability result, where $\mathbf{v}_d(\mathcal A) \in \mathbb R^{\binom{[n]}{d}}$ is the $d$-degree vector of $\mathcal A$ whose $U$-entry is the number of members in $\mathcal A$ containing $U$.
