Common-edge extremal coloring for diagonal Ramsey multiplicity

Construct, for every integer s, a two-coloring of the edges of the complete graph K_{R_2(s)} having exactly m_2(s) monochromatic copies of K_s, with all such copies sharing a common edge.

Background

The paper studies the critical multiplicity m_2(s), defined as the minimum number of monochromatic K_s copies forced in a two-coloring of K_{R_2(s)}. Its extension argument produces colorings of K_{R_2(s)}−e that avoid monochromatic K_s until the final edge is colored.

The authors conjecture that an extremal coloring attaining m_2(s) can always be obtained by coloring such a final edge, and that every forced monochromatic K_s in the resulting coloring contains this same edge. This structural assertion would support the subsequent formula conjecture for m_2(s).