$d$-Degree Erdős-Ko-Rado theorem for finite vector spaces

Abstract

Let $V$ be an $n$-dimensional vector space over the finite field ${\mathbb{F}}_q$ and let ${\left[\genfrac{}{}{0ex}{}{V}{k}\right]}_q$ denote the family of all $k$-dimensional subspaces of $V$. A family $\mathcal{F}\subseteq {\left[\genfrac{}{}{0ex}{}{V}{k}\right]}_q$ is called intersecting if for all $F$, $F^{\prime }\in \mathcal{F}$, we have $\dim$$(F\cap F^{\prime })\ge 1$. Let $\hat{I}{´}_d(\mathcal{F})$ denote the minimum degree in $\mathcal{F}$ of all $d$-dimensional subspaces. In this paper we show that $\hat{I}{´}_d(\mathcal{F})\le \left[\genfrac{}{}{0ex}{}{n-d-1}{k-d-1}\right]$ in any intersecting family $\mathcal{F}\subseteq {\left[\genfrac{}{}{0ex}{}{V}{k}\right]}_q$, where $k>d\ge 2$ and $n\ge 2k+1$.

17 pages