Bounds on the Critical Multiplicity of Ramsey Numbers with Many Colors

The Ramsey number $R(s,t)$ is the least integer $n$ such that any coloring of the edges of $K_n$ with two colors produces either a monochromatic $K_s$ in one color or a monochromatic $K_t$ in the other. If $s=t$, we say that the Ramsey number $R(s,s)$ is diagonal. The critical multiplicity of a diagonal Ramsey number $R(s,s)$, denoted $m(s,s)$ or $m_2(s)$, is the smallest number of copies of a monochromatic $K_s$ that can be found in any coloring of the edges of $K_{R(s,s)}$. For instance, $m_2(2)=1$, $m_2(3)=2$, and $m_2(4)=9$. In this short note, we produce some new upper bounds for the general non-diagonal case of $m(s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_k)$ and improve the bounds on $m_2(s)$ for small $s$. This appears to be the first progress on bounding the critical multiplicity of Ramsey numbers since Piwakowski and Radziszowski’s 2001 determination that $m_2(4)=9$, and we are not aware of any subsequent improvements on this quantity in the literature. We conclude by outlining a reasonably clear path to further improvements.