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 $R(5,5,5) \geq 242$ and $R(6,6,6) \geq 692$. In the second part we extend Fan Chung's result in [C] to show that, $$ R(3,3,3,k_1,k_2,\dots,k_r) \geq 3 R(3,3,k_1,k_2,\dots,k_r) + R(k_1,k_2,\dots,k_r) - 3 $$ holds for any natural number $r$ and for any $k_i\geq 3$, $i=1,2,\dots r$. This general result, along with known results, imply the following nontrivial bounds: $R(3,3,3,4) \geq 91$, $R(3,3,3,5) \geq 137$, $R(3,3,3,6) \geq 165$, $R(3,3,3,7) \geq 220$, $R(3,3,3,9) \geq 336$, and $R(3,3,3,11) \geq 422$.