Bounds on the Critical Multiplicity of Ramsey Numbers with Many Colors

TL;DR

This paper establishes new upper bounds on the critical multiplicity of Ramsey numbers for multiple colors, improving known bounds for small cases and outlining potential for future progress.

Contribution

It introduces novel upper bounds for the critical multiplicity of non-diagonal Ramsey numbers and refines bounds for small diagonal cases, marking progress since 2001.

Findings

New upper bounds for non-diagonal critical multiplicities

Improved bounds for small diagonal cases like m_2(3) and m_2(4)

Outline of a pathway for future improvements

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.