<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$V$</EquationSource> <EquationSource Format="MATHML"><math> <mi>V</mi> </math></EquationSource> </InlineEquation> be an <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-dimensional vector space over the finite field <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{F}_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq4.gif" Format="GIF" Height="36" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$\left[V\atop k\right]_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mfenced close="]" open="["> <mfrac linethickness="0pt"> <mi>V</mi> <mi>k</mi> </mfrac> </mfenced> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> denote the family of all <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$k$</EquationSource> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> </InlineEquation>-dimensional subspaces of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$V$</EquationSource> <EquationSource Format="MATHML"><math> <mi>V</mi> </math></EquationSource> </InlineEquation>. A family <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq7.gif" Format="GIF" Height="36" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal{F}\subseteq \left[V\atop k\right]_q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo>⊆</mo> <msub> <mfenced close="]" open="["> <mfrac linethickness="0pt"> <mi>V</mi> <mi>k</mi> </mfrac> </mfenced> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is called intersecting if for all <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$F$</EquationSource> <EquationSource Format="MATHML"><math> <mi>F</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq9.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$F'\in\mathcal{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo>∈</mo> <mi mathvariant="script">F</mi> </mrow> </math></EquationSource> </InlineEquation>, we have <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="121" /> </InlineMediaObject> <EquationSource Format="TEX">$ \dim (F\cap F')\geq 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>dim</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo>∩</mo> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta_{d}(\mathcal{F})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>δ</mi> <mi>d</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">F</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the minimum degree in <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> of all <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq101.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$d$</EquationSource> <EquationSource Format="MATHML"><math> <mi>d</mi> </math></EquationSource> </InlineEquation>-dimensional subspaces. In this paper we show that <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq14.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="121" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta_{d}(\mathcal{F})\leq \left[ n -d -1\atop k -d -1\right]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>δ</mi> <mi>d</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">F</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mfenced close="]" open="["> <mfrac linethickness="0pt"> <mrow> <mi>n</mi> <mo>-</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>-</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation> in any intersecting family <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq7.gif" Format="GIF" Height="36" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal{F}\subseteq \left[V\atop k\right]_q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo>⊆</mo> <msub> <mfenced close="]" open="["> <mfrac linethickness="0pt"> <mi>V</mi> <mi>k</mi> </mfrac> </mfenced> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq16.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$k>d\geq 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>></mo> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1543_Article_IEq17.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$n\geq 2k+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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

期刊论文
发表日期：

2025年6月20日

查看全文

Y. Shan,
J. Zhou

摘要

Let $V$ $V$ be an $n$ $n$ -dimensional vector space over the finite field $\mathbb{F}_{q}$ $F_{q}$ and let $\left[V\atop k\right]_q$ ${[\binom{V}{k}]}_{q}$ denote the family of all $k$ $k$ -dimensional subspaces of $V$ $V$ . A family $\mathcal{F}\subseteq \left[V\atop k\right]_q$ $F \subseteq {[\binom{V}{k}]}_{q}$ is called intersecting if for all $F$ $F$ , $F'\in\mathcal{F}$ $F^{'} \in F$ , we have $ \dim (F\cap F')\geq 1$ $dim (F \cap F^{'}) \geq 1$ . Let $\delta_{d}(\mathcal{F})$ $δ_{d} (F)$ denote the minimum degree in $\mathcal{F}$ $F$ of all $d$ $d$ -dimensional subspaces. In this paper we show that $\delta_{d}(\mathcal{F})\leq \left[ n -d -1\atop k -d -1\right]$ $δ_{d} (F) \leq [\binom{n - d - 1}{k - d - 1}]$ in any intersecting family $\mathcal{F}\subseteq \left[V\atop k\right]_q$ $F \subseteq {[\binom{V}{k}]}_{q}$ , where $k>d\geq 2$ $k > d \geq 2$ and $n\geq 2k+1$ $n \geq 2 k + 1$ .

\(d\)-degree Erdős-Ko-Rado theorem for finite vector spaces

摘要