Complete-partite extremality for every colour pattern

Prove that, for every k≥3, s≥2, and colour pattern P of K_k, every n-vertex (P,s)-extremal graph is complete partite whenever n is sufficiently large.

Background

Complete multipartite extremality is established in the paper for every non-monochromatic colour pattern, while the general monochromatic case remains connected to unresolved conjectures in the literature. The stated conjecture seeks a uniform structural classification for all individual colour patterns.