Non-trivial $t$-intersecting families for the distance-regular graphs of bilinear forms

TL;DR

This paper characterizes the structure of large non-trivial t-intersecting families of subspaces in a finite vector space, extending classical theorems to the setting of bilinear forms and distance-regular graphs.

Contribution

It extends the Hilton-Milner theorem to non-trivial t-intersecting families in vector spaces over finite fields, describing their structure and maximum size.

Findings

Characterization of maximum size non-trivial t-intersecting families.

Extension of Hilton-Milner theorem to vector space settings.

Structural description of large non-trivial t-intersecting families.

Abstract

Let $V$ be an $(n+\ell)$-dimensional vector space over a finite field, and $W$ a fixed $\ell$-dimensional subspace of $V$. Write ${V\brack n,0}$ to be the set of all $n$-dimensional subspaces $U$ of $V$ satisfying $\dim(U\cap W)=0$. A family $\mathcal{F}\subseteq{V\brack n,0}$ is $t$-intersecting if $\dim(A\cap B)\geq t$ for all $A,B\in\mathcal{F}$. A $t$-intersecting family $\mathcal{F}\subseteq{V\brack n,0}$ is called non-trivial if $\dim(\cap_{F\in\mathcal{F}}F)<t$. In this paper, we describe the structure of non-trivial $t$-intersecting families of ${V\brack n,0}$ with large size. In particular, we show the structure of the non-trivial $t$-intersecting families with maximum size, which extends the Hilton-Milner Theorem for ${V\brack n,0}$.