Determine the asymptotic behaviour of the signed hexagon number

Determine the asymptotic behaviour of the signed hexagon extremal function $R_6(n)=(n,\{C_{-3},C_{+6}\})$, including whether $\limsup_{n\to\infty}R_6(n)/n^{4/3}$ exceeds $2^{1/3}c_6$.

Background

The paper establishes an exact correspondence between R6(n)R_6(n) and C6C_6-free ordinary graphs with an admissible fixed-point-free involution. It constructs signed graphs whose edge density yields the lower coefficient 21/3c6=0.672633550…2^{1/3}c_6=0.672633550\ldots, while the available upper bound comes from the ordinary C6C_6 Turán problem. The authors leave unresolved whether the signed coefficient can exceed the coefficient achieved by their equivariant construction.

References

Determine the asymptotic behaviour of $R_6(n)$. In particular, can $\limsup_{n\to\infty}R_6(n)/n{4/3}$ exceed $2{1/3}c_6$?

— Balanced even cycles in signed graphs:Turán bounds, double covers, and parity obstructions  (2609.10975 - Wang et al., 10 Sep 2026) in Question 1, Section 5, subsection “Consequences and questions”

Determining whether this requirement costs anything in the ordinary hexagon problem remains a natural related question.

— Balanced even cycles in signed graphs:Turán bounds, double covers, and parity obstructions  (2609.10975 - Wang et al., 10 Sep 2026) in Section 5, subsection “Consequences and questions”