AI 中文总结
针对有限域上n维向量空间的子空间r-交叉相交族，证明了当n−k足够大时，其大小乘积的上界及等号成立条件，核心工具为Ellis等人的相交线性映射Junta定理。
AI 中文摘要
设V是有限域GF(q)上的n维向量空间，r≥3，满足(r-1)n≥rk，且F₁,…,Fᵣ⊂[V choose k]，其中[V choose k]表示V的所有k维子空间构成的集合。若对任意Fᵢ∈Fᵢ（1≤i≤r），都有F₁∩…∩Fᵣ≠{0}，则当q和r固定、n−k足够大时，可证∏ᵢ₌₁ʳ|Fᵢ|≤[n−1 choose k−1]，且等号成立当且仅当存在公共直线L，使得每个族Fᵢ由所有包含直线L的k维子空间组成。证明的主要工具之一是Ellis、Kindler和Lifshitz得到的关于相交线性映射的 Junta 定理。
英文摘要
Let $V$ be an $n$-dimensional vector space over a finite field of order $q$. Let $r\geq 3$, $(r-1)n\geq rk$ and let $\mathcal F_1,\ldots,\mathcal F_r\subset \genfrac{[}{]}{0pt}{}{V}{k}$, where $\genfrac{[}{]}{0pt}{}{V}{k}$ denotes the set of $k$-dimensional subspaces of $V$. Suppose that $F_1\cap\cdots\cap F_r\neq\{0\}$ holds for all $F_i\in\mathcal F_i$, $1\leq i\leq r$. Then we show that $\prod_{i=1}^r|\mathcal F_i|\leq\genfrac{[}{]}{0pt}{}{n-1}{k-1}$, provided $n-k$ is sufficiently large for fixed $q$ and $r$. Moreover, equality holds if and only if there is a common line $L$ such that every family $\mathcal F_i$ consists of all $k$-dimensional subspaces containing the line $L$. One of the main tools of the proof is a junta theorem concerning intersecting linear maps obtained by Ellis, Kindler, and Lifshitz.
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