Complete multipartite extremality in the monochromatic Erdős–Rothschild problem

Determine whether, for every pair of integers k≥3 and s≥2 and all sufficiently large n, every extremal graph maximizing the number of s-edge-colourings without a monochromatic copy of K_k is complete multipartite.

Background

The paper reviews known exact and asymptotic cases of the monochromatic Erdős–Rothschild problem. In the known exact cases, the extremal graphs are Turán graphs and therefore complete multipartite, but the general structure of extremal graphs remains unresolved. The question is significant because complete multipartite structure would reduce the graph problem to a finite-dimensional optimization problem over multipartite templates.

References

It is not known whether for all pairs $(k,s)$ every extremal graph is complete multipartite.

— 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 1.1, 'The monochromatic Erdős–Rothschild problem'