$d$-Degree Erd\H{o}s-Ko-Rado theorem for finite vector spaces

TL;DR

This paper extends the Erd ext{"o}s-Ko-Rado theorem to finite vector spaces by establishing an upper bound on the minimum degree of $d$-dimensional subspaces within intersecting families, generalizing classical combinatorial results.

Contribution

It introduces a new bound on the minimum degree of $d$-dimensional subspaces in intersecting families of $k$-subspaces over finite fields, generalizing the Erd ext{"o}s-Ko-Rado theorem.

Findings

Established an upper bound on $ ext{min degree}$ for $d$-dimensional subspaces.

Generalized classical intersecting family results to finite vector spaces.

Applicable for $k>d ext{ and } n ext{ satisfying } n extgreater 2k+1$.

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 ${\rm 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 $n\geq 2k+1$.