{"title":"Disproof of a conjecture of Erdös and moser on tournaments","authors":"K.B. Reid , E.T. Parker","doi":"10.1016/S0021-9800(70)80061-8","DOIUrl":null,"url":null,"abstract":"<div><p>Erdös and Moser [1] displayed a tournament of order 7 with no transitive subtournament of order 4 and conjectured for each positive integer <em>k</em> existence of a tournament of order 2<sup><em>k</em>−1</sup>−1 with no transitive subtournament of order <em>k</em>. The conjecture is disproved for <em>k</em>=5. Further, every tournament of order 14 has a transitive subtournament of order 5. Inductively, the conjecture is false for all orders above 5. Existence and uniqueness of a tournament of order 13 having no transitive subtournament of order 5 are shown.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 3","pages":"Pages 225-238"},"PeriodicalIF":0.0000,"publicationDate":"1970-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80061-8","citationCount":"70","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021980070800618","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 70
Abstract
Erdös and Moser [1] displayed a tournament of order 7 with no transitive subtournament of order 4 and conjectured for each positive integer k existence of a tournament of order 2k−1−1 with no transitive subtournament of order k. The conjecture is disproved for k=5. Further, every tournament of order 14 has a transitive subtournament of order 5. Inductively, the conjecture is false for all orders above 5. Existence and uniqueness of a tournament of order 13 having no transitive subtournament of order 5 are shown.
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