Quasirandomness forcing by the tournaments H9 and H16

Prove that the tournaments H_9 and H_{16} force quasirandomness in regular tournaments.

Background

The paper classifies all tournaments on at most five vertices according to whether they force quasirandomness in regular tournaments, except for H_9 and H_{16}. These tournaments are reverses of one another, so establishing the property for either one establishes it for both.

The authors report that computer-assisted flag-algebra calculations support the conjecture, but the resulting positive-semidefinite matrices are too complicated for them to analyze rigorously or convert into clean rational certificates. Consequently, the forcing property of H_9 and H_{16} remains unresolved.

References

We conjecture that they do indeed have this property. Note that it suffices to consider just one of these tournaments, as they can be obtained from one another by reversing all arcs.

— Forcing Quasirandomness in a Regular Tournament  (2501.11675 - Noel et al., 20 Jan 2025) in Conjecture 1 (labelled conj:H9H16), Section Conclusion