{"title":"3 图形的多色拉姆齐数增长率","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":"{\"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}","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
摘要
k-uniform 超图 G 的 q 色拉姆齐数表示为 r(G;q)，它是这样一个最小整数 N，即 N 个顶点上完整 k-uniform 超图边的任何着色都包含 G 的单色副本。对于三角形来说，一个自然问题可以追溯到 1916 年舒尔的研究，即确定固定 G 和 q 趋于无穷大时 r(G; q) 的行为。在本文中，我们研究了 3-uniform 超图的这一问题，并确定了 r(G; q) 作为 q 的函数的塔高。更确切地说，给定一个超图 G，我们确定了 r(G; q) 在 q 中的多项式、指数或双指数行为。
The growth rate of multicolor Ramsey numbers of 3-graphs
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.
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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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