Characterize the two-color critical multiplicity via balanced edge extensions

Characterize the two-color critical multiplicity m_2(s) by proving that it belongs to the set of values obtained from pairs of integers n_1,n_2 satisfying n_1,n_2\geq s, n_1+n_2=R_2(s)+2, and |n_1-n_2|<R(s-1,s)-(s-2), through the formula \(\min_{i\in\{1,2\}}\prod_{j=0}^{s-3}\left\lceil\frac{(n_i-2)-j}{s-2}\right\rceil\); in particular, establish the associated structural claims about extremal colorings that would imply monotonic increase of m_2(s).

Background

The paper studies the critical multiplicity m_2(s), defined as the minimum number of monochromatic copies of K_s forced in a two-coloring of the complete graph K_{R_2(s)}. Its upper bounds are obtained by considering a coloring of K_{R_2(s)}−e that contains no monochromatic K_s and then examining the two possible colors assigned to the missing edge.

The authors conjecture that an extremal coloring can be chosen so that all monochromatic K_s copies share the final edge, that the two associated monochromatic subgraphs attain the maximum copy count from the counting inequality, and that their vertices outside the common edge partition the remaining vertices. These conditions lead to the displayed set-valued formula for m_2(s). The paper notes that this conjecture would also resolve the previously open question of whether m_2(s) is monotonically increasing.

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\big(s-1,s\big)-(s-2) \right}.

Among other things, \Cref{conjecture} would imply that $m_2(s)$ is monotonically increasing, which has been an open question for some time .

— Bounds on the Critical Multiplicity of Ramsey Numbers with Many Colors  (2501.18869 - Christopherson et al., 31 Jan 2025) in Section 3, “Concluding Remarks,” Conjecture 1, equations (conjbound) and (formula theorem)