s-almost t-intersecting families for vector spaces

Let V be a finite dimensional vector space over a finite field, and $F$ a family consisting of k-subspaces of V. The family $F$ is called t-intersecting if $\dim (F\cap F^{\prime })\ge t$ for any $F,F^{\prime }\in F$. We say $F$ is s-almost t-intersecting if $\left|\{F\in F:\dim (F\cap F^{\prime })<t\}\right|\le s$ for any $F^{\prime }\in F$. 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 $(s,t)\ne (1,1)$ and $k\ge t+2$. The results for 1-almost 1-intersecting families provided by Shan and Zhou are generalized.