Kuo, WentangLiu, Yu-RuRibas, SavioZhou, Kevin2023-10-032023-10-032021-12https://doi.org/10.1016/j.disc.2021.112602http://hdl.handle.net/10012/20006The final publication is available at Elsevier via https://doi.org/10.1016/j.disc.2021.112602. © 2021. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In a previous work [5], we developed the shifted Turán sieve method on a bipartite graph and applied it to problems on cycles in tournaments. More precisely, we obtained upper bounds for the number of tournaments which contain a small number of r-cycles. In this paper, we improve our sieve inequality and apply it to obtain an upper bound for the number of bipartite tournaments which contain a number of 2r-cycles far from the average. We also provide the exact bound for the number of tournaments which contain few 3-cycles, using other combinatorial arguments.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/shifted Turan sievetournamentsbipartite tournaments3-cyclesArticle