A t-intersecting Hilton-Milner theorem for vector spaces

Abstract

Let V be an n-dimensional vector space over a q-element field. For an integer t≥2, a family F of k-dimensional subspaces in V is t-intersecting if dim⁡(F1∩F2)≥t for any F1,F2∈F, and non-trivial if dim⁡(⋂F∈FF)≤t−1. In this paper, we determine the maximum sizes of the non-trivial t-intersecting families for n≥2k+2, k≥t+2, and the extremal structures of families with the maximum sizes have also been characterized. Our results extend the well-known Hilton-Milner theorem for vector spaces to the case of t-intersection.