$s$-almost $t$-intersecting families for vector spaces

Abstract: Let $V$ be a finite dimensional vector space over a finite field, and $\mathcal{F}$ a family consisting of $k$-subspaces of $V$. The family $\mathcal{F}$ is called $t$-intersecting if $\dim(F_{1}\cap F_{2})\geq t$ for any $F_{1}, F_{2}\in \mathcal{F}$. We say $\mathcal{F}$ is $s$-almost $t$-intersecting if for each $F\in \mathcal{F}$ there are at most $s$ members $F^{\prime}$ of $\mathcal{F}$ such that $\dim(F\cap F^{\prime})<t$. In this paper, we prove that $s$-almost $t$-intersecting families with maximum size are $t$-intersecting. We also consider $s$-almost $t$-intersecting families which are not $t$-intersecting, and characterize such families with maximum size for $(s,t)\neq(1,1)$. The result for $1$-almost $1$-intersecting families provided by Shan and Zhou is generalized.