具有多重禁止距离的Rn$\mathbb｛R｝^｛n｝$的色数

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematika Pub Date : 2023-05-09 DOI:10.1112/mtk.12197

Eric Naslund

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Erdős asked about the growth rate of the m-distance chromatic number\n\n ","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":"{\"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\":\"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>. 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引用次数: 0

摘要

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

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

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

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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.