Finite obstruction sets for tournaments

Determine whether every tournament H has finitely many minimal H-homomorphism obstructions among digraphs that are directed-3-cycle-free.

Background

The paper proves finite obstruction characterizations for the particular smooth tournaments T_n. It leaves open whether every tournament enjoys a finite set of minimal obstructions under the same directed-3-cycle-free restriction.

This question isolates the possible role of smoothness in the obstruction result and seeks a classification valid for all finite tournaments.

References

Is is true that for every tournament $$ there a finitely many $\vec{}_3$-free minimal obstructions to $\CSP()$? (Compare to Theorem~\ref{thm:TCn-min-obs}).

— Restricted CSPs and F-free Digraph Algorithmics  (2502.17596 - Guzmán-Pro et al., 24 Feb 2025) in Conclusion and outlook, Section 8