{"title":"一个有效容器引理","authors":"J. Balogh, W. Samotij","doi":"10.19086/DA.17354","DOIUrl":null,"url":null,"abstract":"We prove a new, efficient version of the hypergraph container theorems that is suited for hypergraphs with large uniformities. The main novelty is a refined approach to constructing containers that employs simple ideas from high-dimensional convex geometry. The existence of smaller families of containers for independent sets in such hypergraphs, which is guaranteed by the new theorem, allows us to improve upon the best currently known bounds for several problems in extremal graph theory, discrete geometry, and Ramsey theory.","PeriodicalId":37312,"journal":{"name":"Discrete Analysis","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2019-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"11","resultStr":"{\"title\":\"An efficient container lemma\",\"authors\":\"J. Balogh, W. Samotij\",\"doi\":\"10.19086/DA.17354\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove a new, efficient version of the hypergraph container theorems that is suited for hypergraphs with large uniformities. The main novelty is a refined approach to constructing containers that employs simple ideas from high-dimensional convex geometry. The existence of smaller families of containers for independent sets in such hypergraphs, which is guaranteed by the new theorem, allows us to improve upon the best currently known bounds for several problems in extremal graph theory, discrete geometry, and Ramsey theory.\",\"PeriodicalId\":37312,\"journal\":{\"name\":\"Discrete Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2019-10-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"11\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.19086/DA.17354\",\"RegionNum\":3,\"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":"Discrete Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.19086/DA.17354","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
We prove a new, efficient version of the hypergraph container theorems that is suited for hypergraphs with large uniformities. The main novelty is a refined approach to constructing containers that employs simple ideas from high-dimensional convex geometry. The existence of smaller families of containers for independent sets in such hypergraphs, which is guaranteed by the new theorem, allows us to improve upon the best currently known bounds for several problems in extremal graph theory, discrete geometry, and Ramsey theory.
期刊介绍：
Discrete Analysis is a mathematical journal that aims to publish articles that are analytical in flavour but that also have an impact on the study of discrete structures. The areas covered include (all or parts of) harmonic analysis, ergodic theory, topological dynamics, growth in groups, analytic number theory, additive combinatorics, combinatorial number theory, extremal and probabilistic combinatorics, combinatorial geometry, convexity, metric geometry, and theoretical computer science. As a rough guideline, we are looking for papers that are likely to be of genuine interest to the editors of the journal.
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