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 FF be a family of subspaces of V  . Suppose that dim(F∩F′)⩾tdim(F∩F′)⩾t holds for all F,F′∈FF,F′∈F. Then we show that |F|⩽∑k=dn[nk] for n+t=2dn+t=2d, and |F|⩽∑k=d+1n[nk]+[n−1d] for n+t=2d+1n+t=2d+1. We also consider the case when the condition dim(F∩F′)⩾tdim(F∩F′)⩾t is replaced with dim(F∩F′)≠t−1dim(F∩F′)≠t−1.