On the Density of Transitive Tournaments

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 ⩾ 4 is necessarily quasirandom. We present several other characterizations of quasirandom tournaments nicely complementing previously known results and relatively easily following from our proof techniques.