分拆格中的Erdős–Ko–Rado定理与Hilton–Milner定理
Erdős--Ko--Rado and Hilton--Milner Theorems in the Partition Lattice
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中文总结 AI 辅助
本文针对分拆格相关的拟阵平集相交族问题，在不同参数范围证明了Erdős–Ko–Rado定理，确定了最大非平凡相交族并刻画了极值族，推进了Czabarka的分拆-EKR猜想。
中文摘要 AI 辅助
设$M_n=M(K_{n+1})$为完全图的图拟阵，$\boldsymbol{\textit{F}}_k(M_n)$为其秩-$k$平集。我们研究满足对任意$A,B \boldsymbol{\textit{F}}_k(M_n)$均有$\text{rk}(A \boldsymbol{\textit{B}}) \boldsymbol{\textit{t}}$的子集族$\boldsymbol{\textit{A}} \boldsymbol{\textit{F}}_k(M_n)$。当$\boldsymbol{t}=1$时，该问题等价于Czabarka的分拆- EKR猜想，该猜想由P.~L. Erdős和L.~A. Székely首次公开发表。我们在显式线性范围$n+1\boldsymbol{\textit{8}}k$内证明了对应的Erdős–Ko–Rado定理，向猜想的精确范围$n\boldsymbol{\textit{2}}k$推进了常数因子。对每个固定的$\boldsymbol{t}$，我们还在块数$n+1-k$满足阶为$O_t(k^2)$的显式条件下证明了Erdős–Ko–Rado定理，仅对完整$\boldsymbol{t}$-星取等号。我们还确定了显式$O(k^6)$阈值下最大的非平凡相交族，并在同构意义下刻画了唯一的极值族。
英文摘要
Let $M_n=M(K_{n+1})$ be the graphic matroid of the complete graph, and let $\mathcal{F}_k(M_n)$ be its rank-$k$ flats. We study families $\mathcal{A}\subseteq\mathcal{F}_k(M_n)$ satisfying $\mathrm{rk}(A\wedge B)\ge t$ for all $A,B\in\mathcal{A}$. For $t=1$, this problem is exactly equivalent to Czabarka's partition-EKR conjecture, first introduced in print by P.~L. Erdős and L.~A. Székely~\cite{ErdosSzekelyHigher}. We prove the corresponding Erdős--Ko--Rado theorem in the explicit linear range $n+1\ge8k$, giving a constant-factor advance toward the conjectured sharp range $n\ge2k$. For every fixed $t$, we further prove an Erdős--Ko--Rado theorem under an explicit condition of order $O_t(k^2)$ on the block number $n+1-k$, with equality only for a full $t$-star. We also determine the largest nontrivial intersecting families under an explicit $O(k^6)$ threshold and characterize the unique extremal family up to isomorphism.
