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].
From: Dehai Liu [view email]
[v1]
Mon, 15 Jun 2026 12:26:02 UTC (24 KB)
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