{"title":"Cancellative hypergraphs and Steiner triple systems","authors":"Xizhi Liu","doi":"10.1016/j.jctb.2024.03.006","DOIUrl":null,"url":null,"abstract":"<div><p>A triple system is cancellative if it does not contain three distinct sets <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></math></span> such that the symmetric difference of <em>A</em> and <em>B</em> is contained in <em>C</em>. We show that every cancellative triple system <span><math><mi>H</mi></math></span> that satisfies a particular inequality between the sizes of <span><math><mi>H</mi></math></span> and its shadow must be structurally close to the balanced blowup of some Steiner triple system. Our result contains a stability theorem for cancellative triple systems due to Keevash and Mubayi as a special case. It also implies that the boundary of the feasible region of cancellative triple systems has infinitely many local maxima, thus giving the first example showing this phenomenon.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-04-03","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://www.sciencedirect.com/science/article/pii/S0095895624000248","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
Abstract
A triple system is cancellative if it does not contain three distinct sets such that the symmetric difference of A and B is contained in C. We show that every cancellative triple system that satisfies a particular inequality between the sizes of and its shadow must be structurally close to the balanced blowup of some Steiner triple system. Our result contains a stability theorem for cancellative triple systems due to Keevash and Mubayi as a special case. It also implies that the boundary of the feasible region of cancellative triple systems has infinitely many local maxima, thus giving the first example showing this phenomenon.
如果一个三重系统不包含三个不同的集合 A、B、C，且 A 和 B 的对称差包含在 C 中，那么这个三重系统就是可消三重系统。我们证明，每个满足 H 及其阴影大小之间特定不等式的可消三重系统 H 在结构上一定接近于某个斯坦纳三重系统的平衡炸毁。作为特例，我们的结果包含了基瓦什（Keevash）和穆巴伊（Mubayi）提出的可消三重系统稳定性定理。它还意味着可消三重系统可行区域的边界有无限多个局部最大值，从而给出了第一个显示这一现象的例子。
期刊介绍：
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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