Complete multipartite structure of monochromatic extremal graphs

Determine whether, for every pair of integers k and s, every extremal graph for the monochromatic Erdős–Rothschild problem is complete multipartite.

Background

The paper surveys the monochromatic Erdős–Rothschild problem, in which one seeks the n-vertex graph maximizing the number of s-edge-colourings without a monochromatic K_k. Although all known exact results have complete multipartite extremal graphs, the authors note that the general structural characterization is unresolved. The paper later proves the analogous statement for every non-monochromatic colour pattern, but this does not settle the monochromatic cases.