Non-trivial cross-$t$-intersecting families for vector spaces with the maximum sum of sizes

Abstract

Let $V$ be an $n$-dimensional vector space over a finite field. Suppose that $\mathcal{F}$ and $\mathcal{G}$ are non-empty families of $k$-subspaces and $\ell$-subspaces of $V$, respectively. They are said to be cross-$t$-intersecting if $\dim(F\cap G)\geq t$ for any $F\in\mathcal{F}$ and $G\in \mathcal{G}$, and are further called non-trivial if $\dim(\cap_{F\in\mathcal{F}}F)<t$ and $\dim(\cap_{G\in\mathcal{G}}G)<t$. In this paper, we characterize the non-trivial cross-$t$-intersecting families with the maximum sum of sizes. When $t=1$, our result serves as the $q$-analog of the theorems in [9,11].