Seonghyuk Im, Jaehoon Kim, Joonkyung Lee, Abhishek Methuku
{"title":"关于斯坦纳三重系统中肥大树的埃利奥特-罗德尔猜想的证明","authors":"Seonghyuk Im, Jaehoon Kim, Joonkyung Lee, Abhishek Methuku","doi":"10.1017/fms.2024.34","DOIUrl":null,"url":null,"abstract":"Hypertrees are linear hypergraphs where every two vertices are connected by a unique path. Elliott and Rödl conjectured that for any given <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000343_inline1.png\"/> <jats:tex-math> $\\mu>0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, there exists <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000343_inline2.png\"/> <jats:tex-math> $n_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that the following holds. Every <jats:italic>n</jats:italic>-vertex Steiner triple system contains all hypertrees with at most <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000343_inline3.png\"/> <jats:tex-math> $(1-\\mu )n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> vertices whenever <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000343_inline4.png\"/> <jats:tex-math> $n\\geq n_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove this conjecture.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A proof of the Elliott–Rödl conjecture on hypertrees in Steiner triple systems\",\"authors\":\"Seonghyuk Im, Jaehoon Kim, Joonkyung Lee, Abhishek Methuku\",\"doi\":\"10.1017/fms.2024.34\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Hypertrees are linear hypergraphs where every two vertices are connected by a unique path. Elliott and Rödl conjectured that for any given <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S2050509424000343_inline1.png\\\"/> <jats:tex-math> $\\\\mu>0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, there exists <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S2050509424000343_inline2.png\\\"/> <jats:tex-math> $n_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that the following holds. Every <jats:italic>n</jats:italic>-vertex Steiner triple system contains all hypertrees with at most <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S2050509424000343_inline3.png\\\"/> <jats:tex-math> $(1-\\\\mu )n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> vertices whenever <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S2050509424000343_inline4.png\\\"/> <jats:tex-math> $n\\\\geq n_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove this conjecture.\",\"PeriodicalId\":56000,\"journal\":{\"name\":\"Forum of Mathematics Sigma\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-09-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Forum of Mathematics Sigma\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/fms.2024.34\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum of Mathematics Sigma","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/fms.2024.34","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
A proof of the Elliott–Rödl conjecture on hypertrees in Steiner triple systems
Hypertrees are linear hypergraphs where every two vertices are connected by a unique path. Elliott and Rödl conjectured that for any given $\mu>0$ , there exists $n_0$ such that the following holds. Every n-vertex Steiner triple system contains all hypertrees with at most $(1-\mu )n$ vertices whenever $n\geq n_0$ . We prove this conjecture.
期刊介绍：
Forum of Mathematics, Sigma is the open access alternative to the leading specialist mathematics journals. Editorial decisions are made by dedicated clusters of editors concentrated in the following areas: foundations of mathematics, discrete mathematics, algebra, number theory, algebraic and complex geometry, differential geometry and geometric analysis, topology, analysis, probability, differential equations, computational mathematics, applied analysis, mathematical physics, and theoretical computer science. This classification exists to aid the peer review process. Contributions which do not neatly fit within these categories are still welcome.
Forum of Mathematics, Pi and Forum of Mathematics, Sigma are an exciting new development in journal publishing. Together they offer fully open access publication combined with peer-review standards set by an international editorial board of the highest calibre, and all backed by Cambridge University Press and our commitment to quality. Strong research papers from all parts of pure mathematics and related areas will be welcomed. All published papers will be free online to readers in perpetuity.
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