Almost Intersecting Families for Vector Spaces

Abstract

Let V be an n-dimensional vector space over the finite field \({\mathbb {F}}_{q}\) and let \(\left[ \begin{array}{c} V \\ k \end{array}\right] _q\) denote the family of all k-dimensional subspaces of V. A family \({{\mathcal {F}}}\subseteq \left[ \begin{array}{c} V \\ k \end{array}\right] _q\) is called intersecting if for all F, \(F'\in {{\mathcal {F}}},\) we have \({\textrm{dim}}(F\cap F')\ge 1.\) A family \({{\mathcal {F}}}\subseteq \left[ \begin{array}{c} V \\ k \end{array}\right] _q\) is called almost intersecting if for every \(F\in {{\mathcal {F}}}\) there is at most one element \(F'\in {{\mathcal {F}}}\) satisfying \({\textrm{dim}}(F\cap F')=0.\) 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 \(\left[ \begin{array}{c} V \\ k \end{array}\right] _q,\) 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.