On the Density of Transitive Tournaments

Abstract

We prove that for every fixed $k$, the number of occurrences of the transitive tournament $T{r}_k$ 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.

13 pages, 1 figure

References in corpus (1)

• Graph limits and exchangeable random graphs