\(d\)-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 \(\left[V\atop k\right]_q\) denote the family of all \(k\)-dimensional subspaces of \(V\). A family \(\mathcal{F}\subseteq \left[V\atop k\right]_q\) is called intersecting if for all \(F\), \(F'\in\mathcal{F}\), we have \( \dim (F\cap F')\geq 1\). Let \(\delta_{d}(\mathcal{F})\) denote the minimum degree in \(\mathcal{F}\) of all \(d\)-dimensional subspaces. In this paper we show that \(\delta_{d}(\mathcal{F})\leq \left[ n -d -1\atop k -d -1\right]\) in any intersecting family \(\mathcal{F}\subseteq \left[V\atop k\right]_q\), where \(k>d\geq 2\) and \(n\geq 2k+1\).