$r$-cross $t$-intersecting families for vector spaces

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.

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