The Katona theorem for vector spaces

We present a vector space version of Katonaʼs t-intersection theorem (Katona, 1964 [12]). Let V be the n-dimensional vector space over a finite field, 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 [ n k ] for n + t = 2 d , and | F | ⩽ ∑ k = d + 1 n [ n k ] + [ n − 1 d ] for n + t = 2 d + 1 . We also consider the case when the condition dim ( F ∩ F ′ ) ⩾ t is replaced with dim ( F ∩ F ′ ) ≠ t − 1 .