Extend the fixed-density alternating 4-cycle result

Determine, for every red-edge density sigma in [0,1], the maximum asymptotic number of alternating 4-cycles in red-blue complete graphs with red-edge density sigma+o(1), and determine whether quasirandom graphs are extremal for densities below (1+sqrt(2))/4.

Background

The paper determines the fixed-density profile for the non-alternating RRRB-cycle only when the red density is at least (1+sqrt(2))/4, where quasirandom graphs are extremal. The authors explicitly ask whether this density range can be extended and note uncertainty about quasirandom extremizers at smaller densities. They also formulate the related fixed-density optimization problem for alternating 4-cycles.

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 8.1, “The feasible region of red-blue graphs”