A $t$-intersecting Hilton-Milner theorem for vector spaces for $n=2k+1$ and $q\geq3$

Abstract

Let $V$ be an $n$-dimensional vector space over a $q$-element field. In this paper, we determine the maximum sizes of the maxmial non-trivial $t$-intersecting families  and characterize the extremal structures of families with the maximum sizes  for $n=2k+1$,\ $k\geq t+2,t\geq 2$ and $q\geq3$.  Our results extend the well-known Hilton-Milner theorem for vector spaces to the case of $t$-intersection and improve the applicable range of  parameter $n$ to $n\geq 2k+1+\delta_{2,q}$, where $\delta_{2,q}$ denote the Kronecker delta.