On some Rado numbers for generalized arithmetic progressions

The 2-color Rado number for the equation x₁+x₂−2x₃=c, which for each constant we denote by S₁(c), is the least integer, if it exists, such that every 2-coloring, Δ : 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.