The Katona theorem for vector spaces

We present a vector space version of Katona’s t-intersection theorem [11]. Let V be the n-dimensional vector space over GF(q), and let F be a family of subspaces of V . Suppose that dim(F ∩F ′) ≥ t holds for all F,F ′ ∈ F . Then we show that |F | ≤ ∑k=d [n k ] for n+t = 2d, and |F | ≤ ∑k=d+1 [n k ] + [n−1 d ] for n+t = 2d+1. We also consider the case when the condition dim(F ∩F ′) ≥ t is replaced with dim(F ∩F ′) 6= t−1.