New Lower Bounds for Some Multicolored Ramsey Numbers

In this article we use two different methods to find new lower bounds for some multicolored Ramsey numbers. In the first part we use the finite field method used by Greenwood and Gleason [GG] to show that R(5, 5, 5) 242 and R(6, 6, 6) 692. In the second part we extend Fan Chung's result in [C] to show that, R(3, 3, 3, k1, k2, . . . , kr) 3R(3, 3, k1, k2, . . . , kr) + R(k1, k2, . . . , kr) - 3 holds for any natural number r and for any ki 3, i = 1, 2, . . . r. This general result, along with known results, imply the following nontrivial bounds: R(3, 3, 3, 4) 91, R(3, 3, 3, 5) 137, R(3, 3, 3, 6) 165, R(3, 3, 3, 7) 220, R(3, 3, 3, 9) 336, and R(3, 3, 3, 11) 422.

• Type: Journal
• Year: 1999
• Where: COMBINATORICS
• Authors: Aaron Robertson
• Keywords: COMBINATORICS 1999, Fan Chung, Finite Field Method, Multicolored Ramsey Numbers