Non-trivial $t$-intersecting families for vector spaces

TL;DR

This paper characterizes the structure and maximum size of non-trivial t-intersecting families of k-dimensional subspaces in an n-dimensional vector space over a finite field, extending known results for the case t=1.

Contribution

It provides a detailed description of maximal non-trivial t-intersecting families and identifies those with the largest size, generalizing the Hilton-Milner Theorem for vector spaces.

Findings

Characterization of maximal non-trivial t-intersecting families.

Identification of the largest such families.

Extension of Hilton-Milner Theorem to vector spaces.

Abstract

Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$. In this paper we describe the structure of maximal non-trivial $t$-intersecting families of $k$-dimensional subspaces of $V$ with large size. We also determine the non-trivial $t$-intersecting families with maximum size. In the special case when $t=1$ our result gives rise to the well-known Hilton-Milner Theorem for vector spaces.