Characterize the two-color critical multiplicity via balanced extension sizes

Characterize the two-color critical multiplicity $m_2(s)$ by proving that it belongs to the set of values obtained from pairs of integers $n_1,n_2\geq s$ satisfying $n_1+n_2=R_2(s)+2$ and $|n_1-n_2|<R(s-1,s)-(s-2)$, through the formula \[ m_2(s)\in\left\{\min_{i\in\{1,2\}}\left\{\prod_{j=0}^{s-3}\left\lceil\frac{(n_i-2)-j}{s-2}\right\rceil\right\}: n_1,n_2\geq s,\ n_1+n_2=R_2(s)+2,\ |n_1-n_2|<R(s-1,s)-(s-2)\right\}. \]

Background

The paper studies the critical multiplicity $m(s_1,\ldots,s_k)$, namely the number of monochromatic complete subgraphs that every coloring of the edges of the corresponding Ramsey graph must contain. Its principal upper-bound argument extends a monochromatic-clique-free coloring of $K_{R_2(s)-1}$ to a coloring of $K_{R_2(s)}-e$, then analyzes the monochromatic copies of $K_s$ created when the final edge is assigned either color.

The first conjecture proposes that an extremal coloring can be chosen so that all $m_2(s)$ monochromatic copies share a common edge, that the two color-specific subgraphs attain the counting bound, and that their noncommon vertices partition the graph. The displayed set-valued formula formalizes the first two structural expectations and would substantially sharpen the paper's upper bound.