Quasirandom extremality below the established density range

Determine the maximum density of RRRB-cycles in red-blue complete graphs of prescribed red density σ below (1+√2)/4, and establish whether quasirandom red graphs remain extremal throughout the lower-density range.

Background

The paper proves a density-constrained extremal result for RRRB-cycles when the red density is at least (1+√2)/4≈0.60, showing that quasirandom graphs are extremal there. It explicitly leaves open extending the result to smaller red densities and whether the same quasirandom extremizers persist.

References

An obvious open question is to extend the range of $\sigma$. It is not clear whether we would expect quasirandom graphs to be extremal for smaller $\sigma$.

— The semi-inducibility problem  (2501.09842 - Basit et al., 16 Jan 2025) in Section 7.1, “The feasible region of red-blue graphs”