{"title":"Ramsey numbers upon vertex deletion","authors":"Yuval Wigderson","doi":"10.1002/jgt.23093","DOIUrl":null,"url":null,"abstract":"<p>Given a graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>, its Ramsey number <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $r(G)$</annotation>\n </semantics></math> is the minimum <span></span><math>\n <semantics>\n <mrow>\n <mi>N</mi>\n </mrow>\n <annotation> $N$</annotation>\n </semantics></math> so that every two-coloring of <span></span><math>\n <semantics>\n <mrow>\n <mi>E</mi>\n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>K</mi>\n \n <mi>N</mi>\n </msub>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $E({K}_{N})$</annotation>\n </semantics></math> contains a monochromatic copy of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>, the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number by a super-constant factor. One consequence of this result is the following. There exists a family of graphs <span></span><math>\n <semantics>\n <mrow>\n <mo>{</mo>\n \n <msub>\n <mi>G</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>}</mo>\n </mrow>\n <annotation> $\\{{G}_{n}\\}$</annotation>\n </semantics></math> so that in any Ramsey coloring for <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>G</mi>\n \n <mi>n</mi>\n </msub>\n </mrow>\n <annotation> ${G}_{n}$</annotation>\n </semantics></math> (i.e., a coloring of a clique on <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n <mrow>\n <mo>(</mo>\n \n <msub>\n <mi>G</mi>\n \n <mi>n</mi>\n </msub>\n \n <mo>)</mo>\n </mrow>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> $r({G}_{n})-1$</annotation>\n </semantics></math> vertices with no monochromatic copy of <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>G</mi>\n \n <mi>n</mi>\n </msub>\n </mrow>\n <annotation> ${G}_{n}$</annotation>\n </semantics></math>), one of the color classes has density <span></span><math>\n <semantics>\n <mrow>\n <mi>o</mi>\n <mrow>\n <mo>(</mo>\n \n <mn>1</mn>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $o(1)$</annotation>\n </semantics></math>.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 3","pages":"663-675"},"PeriodicalIF":0.9000,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23093","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23093","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
Given a graph , its Ramsey number is the minimum so that every two-coloring of contains a monochromatic copy of . It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from , the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number by a super-constant factor. One consequence of this result is the following. There exists a family of graphs so that in any Ramsey coloring for (i.e., a coloring of a clique on vertices with no monochromatic copy of ), one of the color classes has density .
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 .