New Upper and Lower Bounds on the Rado Numbers

If E is a linear homogenous equation and c a natural then the Rado number $R_c(E)$ is the least N so that any c-coloring of the positive integers from 1 to N contains a monochromatic solution. Rado characterized for which E R_c(E) always exists. The original proof of Rado's theorem gave enormous bounds on R_c(E) (when it existed). In this paper we establish better upper bounds, and some lower bounds, for R_c(E) for some c and E. In the appendix we use some of our theorems, and ideas from a probabilistic SAT solver, to find many new Rado Numbers.