A t-intersecting Hilton-Milner theorem for vector spaces

Let V be an n-dimensional vector space over a q-element field. For an integer t >= 2, a family Fof k-dimensional subspaces in Vis t-intersecting if dim(F-1 boolean AND F-2) >= t for any F-1, F-2 is an element of F, and non-trivial if dim(boolean AND F-F is an element of F) <= 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. (c) 2023 Elsevier Inc. All rights reserved.