{"title":"The growth rate of multicolor Ramsey numbers of 3-graphs","authors":"Domagoj Bradač, Jacob Fox, Benny Sudakov","doi":"10.1007/s40687-024-00463-w","DOIUrl":null,"url":null,"abstract":"<p>The <i>q</i>-color Ramsey number of a <i>k</i>-uniform hypergraph <i>G</i>, denoted <i>r</i>(<i>G</i>; <i>q</i>), is the minimum integer <i>N</i> such that any coloring of the edges of the complete <i>k</i>-uniform hypergraph on <i>N</i> vertices contains a monochromatic copy of <i>G</i>. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior of <i>r</i>(<i>G</i>; <i>q</i>) for fixed <i>G</i> and <i>q</i> tending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height of <i>r</i>(<i>G</i>; <i>q</i>) as a function of <i>q</i>. More precisely, given a hypergraph <i>G</i>, we determine when <i>r</i>(<i>G</i>; <i>q</i>) behaves polynomially, exponentially or double exponentially in <i>q</i>. This answers a question of Axenovich, Gyárfás, Liu and Mubayi.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40687-024-00463-w","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
The q-color Ramsey number of a k-uniform hypergraph G, denoted r(G; q), is the minimum integer N such that any coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic copy of G. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior of r(G; q) for fixed G and q tending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height of r(G; q) as a function of q. More precisely, given a hypergraph G, we determine when r(G; q) behaves polynomially, exponentially or double exponentially in q. This answers a question of Axenovich, Gyárfás, Liu and Mubayi.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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