The Katona theorem for vector spaces

摘要

We present a vector space version of Katona's 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=dn[nk] for n + t = 2d, and |F|≤∑k=d+1n[nk]+[n-1d] for n + t = 2d + 1. We also consider the case when the condition dim(F ∩ ~F ') ≥ t is replaced with dim(F ∩ ~F ') ≠ t - 1.