{"title":"On the Turán Density of Uniform Hypergraphs","authors":"An Chang, Guo-rong Gao","doi":"10.1007/s10255-023-1067-2","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>p, q</i> be two positive integers. The 3-graph <i>F</i>(<i>p, q</i>) is obtained from the complete 3-graph <i>K</i><span>\n <sup>3</sup><sub><i>p</i></sub>\n \n </span> by adding <i>q</i> new vertices and <span>\\(p(_2^q)\\)</span> new edges of the form <i>vxy</i> for which <i>v</i> ∈ <i>V</i>(<i>K</i><span>\n <sup>3</sup><sub><i>p</i></sub>\n \n </span>) and {<i>x, y</i>} are new vertices. It frequently appears in many literatures on the Turán number or Turán density of hypergraphs. In this paper, we first construct a new class of <i>r</i>-graphs which can be regarded as a generalization of the 3-graph <i>F</i>(<i>p, q</i>), and prove that these <i>r</i>-graphs have the same Turán density under some situations. Moreover, we investigate the Turán density of the <i>F</i>(<i>p, q</i>) for small <i>p, q</i> and obtain some new bounds on their Turán densities.</p></div>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"39 3","pages":"638 - 646"},"PeriodicalIF":0.9000,"publicationDate":"2023-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematicae Applicatae Sinica, English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10255-023-1067-2","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Let p, q be two positive integers. The 3-graph F(p, q) is obtained from the complete 3-graph K3p by adding q new vertices and \(p(_2^q)\) new edges of the form vxy for which v ∈ V(K3p) and {x, y} are new vertices. It frequently appears in many literatures on the Turán number or Turán density of hypergraphs. In this paper, we first construct a new class of r-graphs which can be regarded as a generalization of the 3-graph F(p, q), and prove that these r-graphs have the same Turán density under some situations. Moreover, we investigate the Turán density of the F(p, q) for small p, q and obtain some new bounds on their Turán densities.
期刊介绍:
Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.