Characterize the two-color critical multiplicity via balanced extension colorings

Determine whether, for every integer s, the two-color critical multiplicity m_2(s) belongs to the set of values obtained by minimizing the product of the specified ceiling factors over integers n_1,n_2 satisfying n_1,n_2\ge s, n_1+n_2=R_2(s)+2, and |n_1-n_2|<R(s-1,s)-(s-2).

Background

The paper studies the critical multiplicity m(s_1,\ldots,s_k), namely the guaranteed number of monochromatic complete subgraphs at the corresponding Ramsey threshold. Its main results derive upper bounds by extending a monochromatic-clique-free coloring of K_{R(s_1,\ldots,s_k)-1} to K_{R(s_1,\ldots,s_k)} and counting cliques containing the final edge.

In the two-color diagonal case, the authors conjecture that an extremal coloring can be arranged so that all m_2(s) monochromatic copies of K_s share a common edge, that the two color-induced portions attain the counting bound, and that their vertex sets partition all vertices except for the shared edge. Replacing the relevant inequalities in the preceding upper-bound theorem by equalities yields the displayed conjectural characterization.

References

The first two items seem reasonable, and would yield the following messy-looking conjecture (which is just \Cref{formula theorem} with one inequality replaced by an equality and another by an inclusion):

m_2(s) \in \left{\min_{i \in {1,2}}\left{\prod_{j=0}{s-3}\left\lceil\frac{\big(n_i-2\big)-j}{s-2}\right\rceil\right} : n_1,n_2 \geq s,\ n_1+n_2=R_2(s)+2,\ |n_1-n_2|<R(s-1,s)-(s-2)\right}.

— Bounds on the Critical Multiplicity of Ramsey Numbers with Many Colors  (2501.18869 - Christopherson et al., 31 Jan 2025) in Concluding Remarks, Conjecture 1, equations (conjbound) and (formula theorem)