Almost t-intersecting families for vector spaces

Let V be a finite dimensional vector space over a finite field. A family F consisting of k-subspcaes of V is called almost t-intersecting if for each F∈ F there is at most one F^∈ F with (F∩ F^)≤ t-1. In this paper, we determine the maximum value and the extremal structure of almost t-intersecting families. We also solve the same problems for almost t-intersecting families but not t-intersecting.