C10 Has Positive Turán Density in the Hypercube

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-12-30 DOI:10.1002/jgt.23217

Alexandr Grebennikov, João Pedro Marciano

{"title":"C10 Has Positive Turán Density in the Hypercube","authors":"Alexandr Grebennikov, João Pedro Marciano","doi":"10.1002/jgt.23217","DOIUrl":null,"url":null,"abstract":"<div>\n \n The <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0001\" wiley:location=\"equation/jgt23217-math-0001.png\"><mrow><mrow><mi>n</mi></mrow></mrow></math></annotation>\n </semantics></math>-dimensional hypercube <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>Q</mi>\n \n <mi>n</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0002\" wiley:location=\"equation/jgt23217-math-0002.png\"><mrow><mrow><msub><mi>Q</mi><mi>n</mi></msub></mrow></mrow></math></annotation>\n </semantics></math> is a graph with vertex set <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mn>0</mn>\n \n <mo>,</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n \n <mi>n</mi>\n </msup>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0003\" wiley:location=\"equation/jgt23217-math-0003.png\"><mrow><mrow><msup><mrow><mo class=\"MathClass-open\">{</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo class=\"MathClass-close\">}</mo></mrow><mi>n</mi></msup></mrow></mrow></math></annotation>\n </semantics></math> such that there is an edge between two vertices if and only if they differ in exactly one coordinate. For any graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0004\" wiley:location=\"equation/jgt23217-math-0004.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>, define <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mstyle>\n <mspace></mspace>\n \n <mtext>ex</mtext>\n <mspace></mspace>\n </mstyle>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msub>\n <mi>Q</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>,</mo>\n \n <mi>H</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0005\" wiley:location=\"equation/jgt23217-math-0005.png\"><mrow><mrow><mstyle><mspace width=\"0.1em\"/><mtext>ex</mtext><mspace width=\"0.1em\"/></mstyle><mrow><mo>(</mo><mrow><msub><mi>Q</mi><mi>n</mi></msub><mo>,</mo><mi>H</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> to be the maximum number of edges of a subgraph of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>Q</mi>\n \n <mi>n</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0006\" wiley:location=\"equation/jgt23217-math-0006.png\"><mrow><mrow><msub><mi>Q</mi><mi>n</mi></msub></mrow></mrow></math></annotation>\n </semantics></math> without a copy of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0007\" wiley:location=\"equation/jgt23217-math-0007.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>. In this short note, we prove that for any <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n \n <mo>∈</mo>\n \n <mi>N</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0008\" wiley:location=\"equation/jgt23217-math-0008.png\"><mrow><mrow><mi>n</mi><mo>\\unicode{x02208}</mo><mi mathvariant=\"double-struck\">N</mi></mrow></mrow></math></annotation>\n </semantics></math>,\n\n <div><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mspace></mspace>\n \n <mtext>ex</mtext>\n <mspace></mspace>\n \n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>Q</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>C</mi>\n \n <mn>10</mn>\n </msub>\n \n <mo>)</mo>\n </mrow>\n \n <mo>></mo>\n \n <mn>0.024</mn>\n \n <mo>⋅</mo>\n \n <mi>e</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>Q</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>)</mo>\n </mrow>\n \n <mo>,</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0009\" display=\"block\" wiley:location=\"equation/jgt23217-math-0009.png\"><mrow><mrow><mspace width=\"0.1em\"/><mtext>ex</mtext><mspace width=\"0.1em\"/><mrow><mo>(</mo><msub><mi>Q</mi><mi>n</mi></msub><mo>,</mo><msub><mi>C</mi><mn>10</mn></msub><mo>)</mo></mrow><mo>\\unicode{x0003E}</mo><mn>0.024</mn><mo>\\unicode{x022C5}</mo><mi>e</mi><mrow><mo>(</mo><msub><mi>Q</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>,</mo></mrow></mrow></math></annotation>\n </semantics></math></div> where <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>e</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>Q</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0010\" wiley:location=\"equation/jgt23217-math-0010.png\"><mrow><mrow><mi>e</mi><mrow><mo>(</mo><msub><mi>Q</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> is the number of edges of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>Q</mi>\n \n <mi>n</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23217:jgt23217-math-0011\" wiley:location=\"equation/jgt23217-math-0011.png\"><mrow><mrow><msub><mi>Q</mi><mi>n</mi></msub></mrow></mrow></math></annotation>\n </semantics></math>. Our construction is strongly inspired by the recent breakthrough work of Ellis, Ivan, and Leader, who showed that ‘daisy’ hypergraphs have positive Turán density with an extremely clever and simple linear-algebraic argument.\n </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"31-34"},"PeriodicalIF":0.9000,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23217","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

The $n$ -dimensional hypercube $Q_{n}$ is a graph with vertex set ${0, 1}^{n}$ such that there is an edge between two vertices if and only if they differ in exactly one coordinate. For any graph $H$ , define $ex (Q_{n}, H)$ to be the maximum number of edges of a subgraph of $Q_{n}$ without a copy of $H$ . In this short note, we prove that for any $n \in N$ ,

ex (Q_{n}, C_{10}) > 0.024 \cdot e (Q_{n}),

where

e (Q_{n})

is the number of edges of

Q_{n}

. Our construction is strongly inspired by the recent breakthrough work of Ellis, Ivan, and Leader, who showed that ‘daisy’ hypergraphs have positive Turán density with an extremely clever and simple linear-algebraic argument.

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .