Four Color Rado Numbers for the Equation x₁+x₂+c=x₃

An equation L is said to have a finite 4-color Rado number if there exists a least integer n, such that every 4-coloring of the integers {J I 1 ≤ j ≤ n} has an ordered triple (x1 ,x2 ,x3 ) such that (x1 ,x2 ,x3 ) is a monochromatic solution to L. A particular pattern, the stubborn pattern, is used to provide some framework in the discussion. This pattern is used to prove one of the two general theorems creating upper and lower bounds on the Rado number for the equation x1 + x2 + c = x3 where c is any nonnegative integer. We represent the Rado number for the equation x1 + x2 + c = x3 by Rt (c) where tis the number of colors used and c is the constant c in the equation. An exhaustive computer search is performed to obtain exact 4-color Rado numbers for 3 integers, specifically, R4 (1) = 83, R4 (2) = 121, and R4 (3) = 161. This implies exact values for half of the positive integers, specifically, for all c = 2 (mod3) and c = 3 (mod4). Upper and lower bounds are presented and a conjecture is made regarding the remaining integers