No additional tournaments are quasirandom-forcing

Robert Hancock, Adam Kabela, Daniel Král', Taísa Martins, Roberto Parente, Fiona Skerman, Jan Volec

Abstract: A tournament H<math><mi is="true">H</mi></math> is quasirandom-forcing if the following holds for every sequence (Gn)n∈N<math><msub is="true"><mrow is="true"><mrow is="true"><mo is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub><mo is="true">)</mo></mrow></mrow><mrow is="true"><mi is="true">n</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">N</mi></mrow></msub></math> of tournaments of growing orders: if the density of H<math><mi is="true">H</mi></math> in Gn<math><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math> converges to the expected density of H<math><mi is="true">H</mi></math> in a random tournament, then (Gn)n∈N<math><msub is="true"><mrow is="true"><mrow is="true"><mo is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub><mo is="true">)</mo></mrow></mrow><mrow is="true"><mi is="true">n</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">N</mi></mrow></msub></math> is quasirandom. Every transitive tournament with at least 4 vertices is quasirandom-forcing, and Coregliano (2019) 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 Bucić (2021) that the non-transitive tournaments with seven or more vertices do not have this property.