Kuo, WentangLiu, Yu-RuRibas, SavioZhou, Kevin2023-10-032023-10-032019http://dx.doi.org/10.4153/S000843951900016Xhttp://hdl.handle.net/10012/20005This article has been published in a revised form in the Canadian Mathematical Bulletin http://dx.doi.org/10.4153/S000843951900016X. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. Copyright © Canadian Mathematical Society 2019.Abstract. We construct a shi ed version of the Turán sieve method developed by R. Murty and the second author and apply it to counting problems on tournaments. More precisely, we obtain upper bounds for the number of tournaments which contain a fixed number of restricted r-cycles. These are the first concrete results which count the number of cycles over “all tournaments”.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/shifted Turan sievetournamentcycleArticle