The pigenhole principle and multicolor Ramsey numbers

TL;DR

This paper derives explicit upper bounds for multicolor Ramsey numbers using the pigeonhole principle, improving previous bounds and analyzing secondary terms to refine asymptotic estimates.

Contribution

It provides a new, self-contained proof of upper bounds for $R_r(k)$ based solely on the pigeonhole principle, including secondary term analysis and improved constants.

Findings

Improved explicit bounds for $R_r(k)$ for $r ext{ } ext{and} ext{ }k ext{ } ext{large}$

Asymptotic formula for $R_r(k)$ with a better constant factor

Comparison of bounds with numerical data

Abstract

For integers $k,r\geq 2$, 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 on 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 $r\geq 3$, and we also consider an often ignored secondary term, which allows us to subtract a uniformly bounded below positive proportion of the main term. Asymptotically, we give a self-contained proof that $R_r(k)\leq \left(\frac{3+e}{2}\right)\frac{(r(k-2))!}{((k-2)!)^r}(1+o_{r\to \infty}(1)),$ and we conclude by noting that our methods combine with previous estimates on $R_r(3)$ to improve the constant $\frac{3+e}{2}$ to $\frac{3+e}{2}-\frac{d}{48}$, where $d=66-R_4(3)\geq 4$. We also compare our formulas, and previously documented formulas, to some collected numerical data.