On the Density of Transitive Tournaments

TL;DR

This paper investigates the minimal density of transitive tournaments within larger tournaments, establishing that random tournaments asymptotically minimize this density and characterizing sequences achieving this minimum as quasi-random.

Contribution

It proves that random tournaments minimize transitive tournament occurrences and characterizes sequences achieving this minimum as quasi-random, providing new insights into tournament structure.

Findings

Random tournaments asymptotically minimize transitive tournament density.

Sequences achieving minimum density are necessarily quasi-random.

Multiple new characterizations of quasi-random tournaments are provided.

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 $k\geq 4$ 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.