Structural characterization of extremal two-colorings

Establish that, for every integer s, an extremal two-coloring of K_{R_2(s)} attaining m_2(s) can be chosen so that the two monochromatic subgraphs generated by the final edge each attain the maximum number of K_s copies allowed by the counting inequality, and their vertex sets partition all vertices except for the two endpoints of the common edge.

Background

The counting inequality bounds the number of K_s copies in a graph when all copies share a distinguished edge and deleting that edge leaves no K_s. The paper applies this inequality separately to the two colors arising when a K_{R_2(s)}−e coloring is extended across its final edge.

The conjectured structure asserts simultaneously that both color classes achieve the counting bound and that no vertices outside the two common endpoints are shared between the corresponding monochromatic subgraphs. In particular, their orders would satisfy |V(G_1)|+|V(G_2)|=R_2(s)+2.