No additional tournaments are quasirandom-forcing

TL;DR

This paper proves that aside from certain small transitive and a specific non-transitive 5-vertex tournament, no other tournaments possess the quasirandom-forcing property, extending previous classifications.

Contribution

It establishes that no additional tournaments beyond known cases are quasirandom-forcing, completing the classification for tournaments with more than four vertices.

Findings

Only known small transitive tournaments are quasirandom-forcing.

A unique non-transitive 5-vertex tournament is quasirandom-forcing.

No larger tournaments beyond these have the quasirandom-forcing property.

Abstract

A tournament H is quasirandom-forcing if the following holds for every sequence (G_n) of tournaments of growing orders: if the density of H in G_n converges to the expected density of H in a random tournament, then (G_n) is quasirandom. Every transitive tournament with at least 4 vertices is quasirandom-forcing, and Coregliano et al. [Electron. J. Combin. 26 (2019), P1.44] showed that there is also a non-transitive 5-vertex tournament with the property. We show that no additional tournament has this property. This extends the result of Bucic et al. [Combinatorica 41 (2021), 175-208] that the non-transitive tournaments with seven or more vertices do not have this property.