The pigeonhole principle and multicolor Ramsey numbers

Abstract

For integers $k,r\ge 2$, the diagonal Ramsey number $R_r(k)$ is the minimum $N\in \mathbb{N}$ such that every $r$-coloring of the edges of a complete graph on $N$ vertices yields a monochromatic subgraph on $k$ vertices. Here we make a careful effort of extracting explicit upper bounds for $R_r(k)$ from the pigeonhole principle alone. Our main term improves on previously documented explicit bounds for $r\ge 3$, and we also consider an often-ignored secondary term, which allows us to subtract a positive proportion of the main term that is uniformly bounded below. Asymptotically, we give a self-contained proof that

$$R_r(k)\le \left(\frac{3+e}{2}\right)\frac{(r(k-2))!}{{((k-2)!)}^r}(1+o_{r\to \infty }(1)),$$

and we conclude by noting that our methods combine with previous estimates on $R_r(3)$ to improve the constant $\frac{1}{2}(3+e)$ to $\frac{1}{2}(3+e)-\frac{1}{48}d$, where $d=66-R_4(3)\ge 4$. We also compare our formulas, and previously documented formulas, to some collected numerical data.