Construct explicit tournaments exhibiting substantial non-additivity

Construct explicit tournaments for which the inversion number of a dijoin is substantially less than the sum of the inversion numbers of its two constituent tournaments, or construct explicit counterexamples to additivity of tournament minimum rank.

Background

The paper proves non-constructively that additivity of inversion numbers for dijoins can fail by an arbitrarily large factor, and that tournament minimum rank can likewise be far from additive. The existence proofs select suitable symmetric matrices by counting arguments rather than providing a direct construction.

The authors emphasize that no general method is currently available for constructing explicit tournaments with large inversion numbers. Consequently, an explicit family realizing substantial failure of inversion additivity—or even an explicit counterexample to additivity of tournament minimum rank—remains unresolved.

References

Finding explicit tournaments for which additivity fails badly, or even explicit counterexamples to additivity of tournament minimum rank, remains an interesting and open problem.

— On the failure of additivity of inversion numbers for dijoins  (2610.13798 - Behague et al., 9 Oct 2026) in Section 5, “Concluding Remarks” (Section 5, concluding paragraph)