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

Yunjing Shan; Junling Zhou

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arXiv · 2411.17985

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

Yunjing Shan·Junling Zhou

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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 $n\geq 2k+1$.

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BibTeX RIS

Yunjing Shan, Junling Zhou. 2024-11-27. $d$-Degree Erd\H{o}s-Ko-Rado theorem for finite vector spaces. https://arxiv.org/abs/2411.17985

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