Extension of stability frameworks to broader forbidden subdigraph classes

Determine whether the quantitative stability frameworks established for $\vec{C}_3$-free digraphs extend to broader classes of forbidden subdigraphs.

Background

The paper proves a quantitative stability theorem for loopless C⃗3\vec{C}_3-free digraphs: digraphs whose squared out-degree norm is close to the extremal value are close in edit distance to the ordered digon-chain construction F⃗n,2\vec{F}_{n,2}.

The conclusion explicitly identifies the extension of these stability methods beyond the directed triangle as unresolved. The paper does not specify which broader forbidden-subdigraph classes admit analogous stability results or what their extremal constructions should be.

References

Future research should investigate whether these stability frameworks extend to broader classes of forbidden subdigraphs and whether the exact result for $Reg_5$ can be formally proved using similar algebraic reductions.

— $L_2$ Turán Problems for Small Tournaments and Stability  (2609.05042 - Iľkovič, 4 Sep 2026) in Section 7, “Conclusions”