If E is a linear homogenous equation and c ∈ N 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.
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