论文标题
$ r $ - 十字$ t $ - 矢量空间的家庭
$r$-cross $t$-intersecting families for vector spaces
论文作者
论文摘要
令$ v $为有限字段$ \ mathbb {f} _q $上的$ n $维矢量空间，而$ {v \ brack k} $表示$ v $的所有$ k $二维子空间的家族。家属$ \ Mathcal {f} _1 \ subseteq {v \ brack k_1}，\ Mathcal {f} _2 _2 \ subseteq {v \ brack k_2}，\ ldots，\ ldots，\ ldots，\ mathcal，\ mathcal {f} _r \ subseteeq $ bers $ bers $ bers $ $t$-intersecting if $\dim(F_1\cap F_2\cap\cdots\cap F_r)\geq t$ for all $F_i\in\mathcal{F}_i,\ 1\leq i\leq r.$ The $r$-cross $t$-intersecting families $\mathcal{F}_1$,如果$ \ dim（\ cap_ {1 \ cap_ {1 \ leq i \ leq r} \ cap_ \ cap_ {f \ in \ mathcal {f} f} f}，$ \ mathcal {f} _2，\ \ ldots，\ mathcal {f} _r $被认为是非琐事的。在本文中，我们首先确定$ r $ - 十字$ t $ t $更新的家庭的结构，其尺寸最高的产品。结果，我们部分证明了Frankl和Tokushige的猜想之一，大约$ r $ $ $ 1 $ 1 $ $ 1 $ - 对矢量空间的家庭。然后，我们描述了非平凡$ r $ -r $ -cross $ t $ - 缩减家庭$ \ Mathcal {f} _1 $，$ \ MATHCAL {f} _2，\ ldots，\ Mathcal {f} _r $ \ MATHCAL {F} _1 = \ MATHCAL {F} _2 = \ CDOTS = \ Mathcal {f} _r = \ Mathcal {F} $，其中$ \ Mathcal {f} $在下一个假设中是$ r $ r $ - r $ - t $ - t $ t $ - t $ - t $ - t $ - t $ t $ - 同时，还证明了非平凡$ r $ r $ t $ t $ distecting家庭的稳定性结果。
Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F}_1\subseteq{V\brack k_1},\mathcal{F}_2\subseteq{V\brack k_2},\ldots,\mathcal{F}_r\subseteq{V\brack k_r}$ are said to be $r$-cross $t$-intersecting if $\dim(F_1\cap F_2\cap\cdots\cap F_r)\geq t$ for all $F_i\in\mathcal{F}_i,\ 1\leq i\leq r.$ The $r$-cross $t$-intersecting families $\mathcal{F}_1$, $\mathcal{F}_2,\ldots,\mathcal{F}_r$ are said to be non-trivial if $\dim(\cap_{1\leq i\leq r}\cap_{F\in\mathcal{F}_i}F)<t$. In this paper, we first determine the structure of $r$-cross $t$-intersecting families with maximum product of their sizes. As a consequence, we partially prove one of Frankl and Tokushige's conjectures about $r$-cross $1$-intersecting families for vector spaces. Then we describe the structure of non-trivial $r$-cross $t$-intersecting families $\mathcal{F}_1$, $\mathcal{F}_2,\ldots,\mathcal{F}_r$ with maximum product of their sizes under the assumptions $r=2$ and $\mathcal{F}_1=\mathcal{F}_2=\cdots=\mathcal{F}_r=\mathcal{F}$, respectively, where the $\mathcal{F}$ in the latter assumption is well known as $r$-wise $t$-intersecting family. Meanwhile, stability results for non-trivial $r$-wise $t$-intersecting families are also been proved.
