{"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":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Research in the Mathematical Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40687-024-00463-w","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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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.
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Research in the Mathematical Sciences is an international, peer-reviewed hybrid journal covering the full scope of Theoretical Mathematics, Applied Mathematics, and Theoretical Computer Science. The Mission of the Journal is to publish high-quality original articles that make a significant contribution to the research areas of both theoretical and applied mathematics and theoretical computer science.
This journal is an efficient enterprise where the editors play a central role in soliciting the best research papers, and where editorial decisions are reached in a timely fashion. Research in the Mathematical Sciences does not have a length restriction and encourages the submission of longer articles in which more complex and detailed analysis and proofing of theorems is required. It also publishes shorter research communications (Letters) covering nascent research in some of the hottest areas of mathematical research. This journal will publish the highest quality papers in all of the traditional areas of applied and theoretical areas of mathematics and computer science, and it will actively seek to publish seminal papers in the most emerging and interdisciplinary areas in all of the mathematical sciences. Research in the Mathematical Sciences wishes to lead the way by promoting the highest quality research of this type.
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