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This article is cited in 17 scientific papers (total in 17 papers)
On the density of transitive tournaments
L. N. Coreglianoa, A. A. Razborovb a Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP, Brazil
b University of Chicago, Chicago, Illinois 60637
Abstract:
We prove that for every fixed $k$, the number of occurrences of the transitive tournament $Tr_k$ of order $k$ in a tournament $T_n$ on $n$ vertices is asymptotically minimized when $T_n$ is random. In the opposite direction, we show that any sequence of tournaments $\{T_n\}$ achieving this minimum for any fixed $k\ge 4$ is necessarily quasi-random. We present several other characterizations of quasi-random tournaments nicely complementing previously known results and relatively easily following from our proof techniques.
Received: 10.02.2015 Revised: 11.03.2016
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