The chromatic number of R n $\mathbb {R}^{n}$ with multiple forbidden distances

IF 0.8 3区数学 Q2 MATHEMATICS Mathematika Pub Date : 2023-05-09 DOI:10.1112/mtk.12197

Eric Naslund

{"title":"The chromatic number of \n \n \n R\n n\n \n $\\mathbb {R}^{n}$\n with multiple forbidden distances","authors":"Eric Naslund","doi":"10.1112/mtk.12197","DOIUrl":null,"url":null,"abstract":"<p>Let <math>\n <semantics>\n <mrow>\n <mi>A</mi>\n <mo>⊂</mo>\n <msub>\n <mi>R</mi>\n <mrow>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n </msub>\n </mrow>\n <annotation>$A\\subset \\mathbb {R}_{>0}$</annotation>\n </semantics></math> be a finite set of distances, and let <math>\n <semantics>\n <mrow>\n <msub>\n <mi>G</mi>\n <mi>A</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$G_{A}(\\mathbb {R}^{n})$</annotation>\n </semantics></math> be the graph with vertex set <math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {R}^{n}$</annotation>\n </semantics></math> and edge set <math>\n <semantics>\n <mrow>\n <mo>{</mo>\n <mrow>\n <mo>(</mo>\n <mi>x</mi>\n <mo>,</mo>\n <mi>y</mi>\n <mo>)</mo>\n </mrow>\n <mo>∈</mo>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <mo>:</mo>\n <mspace></mspace>\n <mo>∥</mo>\n <mi>x</mi>\n <mo>−</mo>\n <mi>y</mi>\n <msub>\n <mo>∥</mo>\n <mn>2</mn>\n </msub>\n <mo>∈</mo>\n <mi>A</mi>\n <mo>}</mo>\n </mrow>\n <annotation>$\\lbrace (x,y)\\in \\mathbb {R}^{n}:\\ \\Vert x-y\\Vert _{2}\\in A\\rbrace$</annotation>\n </semantics></math>, and let <math>\n <semantics>\n <mrow>\n <mi>χ</mi>\n <mrow>\n <mo>(</mo>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <mo>,</mo>\n <mi>A</mi>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <mi>χ</mi>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>G</mi>\n <mi>A</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <mo>)</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\chi (\\mathbb {R}^{n},A)=\\chi (G_{A}(\\mathbb {R}^{n}))$</annotation>\n </semantics></math>. Erdős asked about the growth rate of the <i>m</i>-distance chromatic number\n\n </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12197","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Let $A \subset R_{> 0}$ be a finite set of distances, and let $G_{A} (R^{n})$ be the graph with vertex set $R^{n}$ and edge set ${(x, y) \in R^{n} : ∥ x - y ∥_{2} \in A}$ , and let $χ (R^{n}, A) = χ (G_{A} (R^{n}))$ . Erdős asked about the growth rate of the m-distance chromatic number

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

具有多重禁止距离的Rn$\mathbb｛R｝^｛n｝$的色数

设A⊂R>；0$A\subet\mathbb{R}_{>；0}$是一组有限的距离，设G A（Rn）$G_｛A｝（\mathbb｛R｝^｛n｝）$为具有顶点集的图Rn$\mathbb｛R｝^｛n｝$和边集｛（x，y）∈Rn:∈x−y∈2∈A}$\lbrace（x，y）\in\mathbb｛R｝^｛n｝：\\Vert x-y\Vert _｛2｝\ in A\rbrace$，设χ（Rn，A）=χ（G A（R n））$\chi（\mathbb｛R｝^｛n｝，A）=\chi（G_｛A｝（\mathbb｛R｝^｛n｝））$。Erdõs询问m距离色数的增长率

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Mathematika MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

1.40

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.

期刊最新文献

Twisted mixed moments of the Riemann zeta function Diophantine approximation by rational numbers of certain parity types Issue Information The local solubility for homogeneous polynomials with random coefficients over thin sets A discrete mean value of the Riemann zeta function