关于 Frankl--Tokushige 猜想及向量空间中几乎完全的 $r$-交叉 $t$-相交定理

中文摘要 AI 辅助

设 $r\geq3$ 且 $k_1\geq k_2\geq\cdots\geq k_r\geq t$。令 $\mathcal{F}_1,\mathcal{F}_2,\ldots,\mathcal{F}_r$ 为有限域 $\mathbb{F}_q$ 上 $n$ 维向量空间中的子空间族，其维数分别为 $k_1,k_2,\ldots,k_r$。若对于所有 $F_i \in \mathcal{F}_i$（$i = 1,2,\dots,r$），均有 $\dim \left(F_{1} \cap F_{2} \cap \cdots \cap F_{r}\right) \geq t$，则称这 $r$ 个族为 $r$-交叉 $t$-相交的。2016年，Frankl 和 Tokushige 猜想：当 $t=1$ 且 $n\geq rk_1/(r-1)$ 时，$\prod_{i=1}^{r}|\mathcal{F}_i|\leq\prod_{i=1}^{r}{n-1\brack k_i-1}$。这一引人注目的猜想建议建立 $n\sim ck_1$（其中 $c=c(r)\in(1,2)$）情形的相交定理，该方向长期以来一直具有挑战性。在本文中，我们克服了这一障碍，证明了：对于所有 $t\geq1$ 且 $n\geq rk_1/(r-1)+C(t,r)$，其中 $C(t,r)=rt/(r-1)+1$，有 $\prod_{i=1}^{r}|\mathcal{F}_i|\leq\prod_{i=1}^{r}{n-t\brack k_i-t}$。这证明了 Frankl--Tokushige 猜想（除至多三个 $n$ 值外），并为几乎所有参数值建立了 Erdős--Ko--Rado 型定理。此外，我们刻画了所有极值配置。我们的证明是纯组合的，基于 $t$-覆盖方法，并进行了若干关键改进。我们还获得了 $r$-重 $t$-相交族和非平凡 $r$-交叉 $t$-相交族的几乎完全相交定理。

英文摘要

Let $r\geq3$ and $k_1\geq k_2\geq\cdots\geq k_r\geq t$. Let $\mathcal{F}_1,\mathcal{F}_2,\ldots,\mathcal{F}_r$ be families of subspaces, of respective dimensions $k_1,k_2,\ldots,k_r$, in an $n$-dimensional vector space over the finite field $\mathbb{F}_q$. The $r$ families are called $r$-cross $t$-intersecting if $\dim \left(F_{1} \cap F_{2} \cap \cdots \cap F_{r}\right) \geq t$ for all $F_{i} \in \mathcal{F}_{i}, i = 1,2,\dots,r$. In 2016, Frankl and Tokushige conjectured that $\prod_{i=1}^{r}|\mathcal{F}_i|\leq\prod_{i=1}^{r}{n-1\brack k_i-1}$ for $t=1$ and $n\geq rk_1/(r-1)$. The appealing conjecture suggests establishing intersection theorems for $n\sim ck_1$ with $c=c(r)\in(1,2)$, a direction that has long been challenging. In this paper, we overcome this barrier by proving that $$\prod_{i=1}^{r}|\mathcal{F}_i|\leq\prod_{i=1}^{r}{n-t\brack k_i-t}\;\;\mbox{for all}\;\;t\geq1\;\mbox{and}\;n\geq rk_1/(r-1)+C(t,r),$$ where $C(t,r)=rt/(r-1)+1$. This proves the Frankl--Tokushige conjecture except for at most three values of $n$, and establishes an Erdős--Ko--Rado type theorem for almost all values of parameters. Furthermore, we characterize all extremal configurations. Our proof is purely combinatorial and based on the $t$-cover method, with several essential refinements. We also obtain almost complete intersection theorems for $r$-wise $t$-intersecting families and non-trivial $r$-cross $t$-intersecting families.

发表机构

• Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院数学与复杂系统教育部重点实验室）

机构由 AI 辅助整理，请以论文原文为准。