Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
The pigeonhole principle and multicolor Ramsey numbers

### Vishal Balaji, Powers Lamb, Andrew Lott, Dhruv Patel, Alex Rice, Sakshi Singh and Christine Rose Ward

Vol. 15 (2022), No. 5, 857–884
##### Abstract

For integers $k,r\ge 2$, the diagonal Ramsey number ${R}_{r}\left(k\right)$ is the minimum $N\in ℕ$ 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 ${R}_{r}\left(k\right)$ 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

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

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

##### Keywords
Ramsey theory, Ramsey number, party problem, pigeonhole principle
Primary: 05C15