The pigeonhole principle and multicolor Ramsey numbers

For integers k; r >= 2, the diagonal Ramsey number R-r(k) is the minimum N is an element of 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 >= 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) <= (3+e/2) (r(k - 2))!/((k - 2)!)(r) (1+ o(r ->infinity) (1)) and we conclude by noting that our methods combine with previous estimates on Rr .3/ to improve the constant 1/2(3 + e( to 1/2(3 + e) - 1/48d, where d = 66 -R-4(3) >= 4. We also compare our formulas, and previously documented formulas, to some collected numerical data.