Almost Intersecting Families for Vector Spaces.

Let V be an n-dimensional vector space over the finite field F q and let V k q denote the family of all k-dimensional subspaces of V. A family F ⊆ V k q is called intersecting if for all F, F ′ ∈ F , we have dim (F ∩ F ′) ≥ 1. A family F ⊆ V k q is called almost intersecting if for every F ∈ F there is at most one element F ′ ∈ F satisfying dim (F ∩ F ′) = 0. In this paper we investigate almost intersecting families in the vector space V. Firstly, for large n, we determine the maximum size of an almost intersecting family in V k q , which is the same as that of an intersecting family. Secondly, we characterize the structures of all maximum almost intersecting families under the condition that they are not intersecting. [ABSTRACT FROM AUTHOR]