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

Abstract

Let $V$ be an $(n+\ell )$-dimensional vector space over a finite field, and $W$ a fixed $\ell$-dimensional subspace of $V$. Write $\left[\genfrac{}{}{0ex}{}{V}{n,0}\right]$ to be the set of all $n$-dimensional subspaces $U$ of $V$ satisfying $\dim (U\cap W)=0$. A family $\mathcal{F}\subseteq \left[\genfrac{}{}{0ex}{}{V}{n,0}\right]$ is $t$-intersecting if $\dim (A\cap B)\ge t$ for all $A,B\in \mathcal{F}$. A $t$-intersecting family $\mathcal{F}\subseteq \left[\genfrac{}{}{0ex}{}{V}{n,0}\right]$ 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 $\left[\genfrac{}{}{0ex}{}{V}{n,0}\right]$ 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 $\left[\genfrac{}{}{0ex}{}{V}{n,0}\right]$.

25 pages

References in corpus (3)

• Non-trivial $t$-intersecting families for vector spaces
• The structure of large non-trivial t-intersecting families for finite sets
• Hilton-Milner results in projective and affine spaces