On some Rado numbers for generalized arithmetic progressions

The 2-color Rado number for the equation x1 + x2 − 2x3 = c, which for each constant c ∈ Z we denote by S1(c), is the least integer, if it exists, such that every 2-coloring, ∆ : [1, S1(c)]→ {0, 1}, of the natural numbers admits a monochromatic solution to x1 +x2−2x3 = c, and otherwise S1(c) = ∞. We determine the 2-color Rado number for the equation x1 + x2 − 2x3 = c, when additional inequality restraints on the variables are added. In particular, the case where we require x2 < x3 < x1, is a generalization of the 3-term arithmetic progression; and the work done here improves previously established upper bounds to an exact value.