Bounds on the Critical Multiplicity of Ramsey Numbers with Many Colors
Bryce Christopherson, Casia Steinhaus
TL;DR
This work studies the critical multiplicity $m(s_1,\dots,s_k)$ of Ramsey numbers, i.e., the minimum guaranteed number of monochromatic $K_{s_i}$'s in any $k$-coloring of $K_{R(s_1,\dots,s_k)}$. The authors combine an extension argument (extending colorings from $K_{n-1}$ to $K_n$ while avoiding prescribed monochromatic subgraphs) with a counting inequality for copies of $K_s$ sharing a common edge to derive a general upper bound: $m(s_1,\dots,s_k) \le |E(G)| \prod_{j=0}^{\max s_i-3} \left\lceil \frac{\frac{R(s_1,\dots,s_k) - |V(G)|}{k} - j}{\min s_t - 2} \right\rceil$, for suitable connected subgraphs $G$ of $K_n$ with $n=R(s_1,\dots,s_k)$. Specializing to $G=K_2$ yields improved bounds for small parameters and extends to off-diagonal and multi-color settings. The paper also provides a refined diagonal bound for $m_2(s)$ using a two-edge extension framework, giving concrete improvements such as $m_2(5) \le 392$. These results constitute the first progress on bounding the critical multiplicity since the 2001 determination that $m_2(4)=9$, and they sketch a clear path for further tightening via carefully constructed extendable colorings.
Abstract
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 $m(s_1,...,s_k)$ 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.