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The Shifted Turán Sieve Method on Tournaments

Part of: Combinatorics Multiplicative number theory

Published online by Cambridge University Press:  11 April 2019

Wentang Kuo ,
Yu-Ru Liu ,
Sávio Ribas  and
Kevin Zhou
Wentang Kuo
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada Email: wtkuo@uwaterloo.cayrliu@uwaterloo.cakkqzhou@gmail.com
Yu-Ru Liu
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada Email: wtkuo@uwaterloo.cayrliu@uwaterloo.cakkqzhou@gmail.com
Sávio Ribas
Affiliation:
Departamento de Matemática, Universidade Federal de Ouro Preto, Ouro Preto, MG, 35400-000, Brazil Email: savio.ribas@ufop.edu.br
Kevin Zhou
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada Email: wtkuo@uwaterloo.cayrliu@uwaterloo.cakkqzhou@gmail.com
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Abstract

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We construct a shifted 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”.

Keywords

shifted Turán sievetournamentcycle

MSC classification

Primary: 05A05: Permutations, words, matrices
Secondary: 11N35: Sieves
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 841 - 855
Copyright
© Canadian Mathematical Society 2019 

Footnotes

The research of the first and second authors were partially supported by NSERC discovery grants. The research of the third author was partially supported by CAPES and CSF/CNPQ, Brazil.

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