{"title":"Turán numbers and switching","authors":"Karen Gunderson , Jason Semeraro","doi":"10.1016/j.disc.2024.114275","DOIUrl":null,"url":null,"abstract":"<div><div>Using a switching operation on tournaments we obtain some new lower bounds on the Turán number of the <em>r</em>-graph on <span><math><mi>r</mi><mo>+</mo><mn>1</mn></math></span> vertices with 3 edges. For <span><math><mi>r</mi><mo>=</mo><mn>4</mn></math></span>, extremal examples were constructed using Paley tournaments in previous work. We show that these examples are unique (in a particular sense) using Fourier analysis.</div><div>A 3-tournament is a ‘higher order’ version of a tournament given by an alternating function on triples of distinct vertices in a vertex set. We show that 3-tournaments also enjoy a switching operation and use this to give a formula for the size of a switching class in terms of level permutations, generalising a result of Babai–Cameron.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114275"},"PeriodicalIF":0.7000,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004060","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Using a switching operation on tournaments we obtain some new lower bounds on the Turán number of the r-graph on vertices with 3 edges. For , extremal examples were constructed using Paley tournaments in previous work. We show that these examples are unique (in a particular sense) using Fourier analysis.
A 3-tournament is a ‘higher order’ version of a tournament given by an alternating function on triples of distinct vertices in a vertex set. We show that 3-tournaments also enjoy a switching operation and use this to give a formula for the size of a switching class in terms of level permutations, generalising a result of Babai–Cameron.
期刊介绍：
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
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