Complete multipartite conjecture for all colour patterns

Prove that, for every k≥3, s≥2, and colour pattern P of K_k, every sufficiently large n-vertex graph maximizing the number of P-free s-edge-colourings is complete multipartite.

Background

Complete multipartite extremal graphs arise throughout the known results for generalized Erdős–Rothschild problems, and the paper proves this structure for all non-monochromatic patterns covered by its framework. The authors propose the universal statement that the same structure holds for every colour pattern, including the difficult unresolved monochromatic cases.

References

We conjecture that for every pattern, every extremal graph for the generalised Erdős-Rothschild problem is complete partite.

— A framework for the generalised Erdős-Rothschild problem and a resolution of the dichromatic triangle case  (2502.12291 - Gupta et al., 17 Feb 2025) in Section 6, subsection “Other colour patterns”