New Proofs for the Disjunctive Rado Number of the Equations $$x_1-x_2=a$$ x 1 - x 2 = a and $$x_1-x_2=b$$ x 1 - x 2 = b

Let m, a, b be positive integers, with $$\gcd (a,b)=1$$ . The disjunctive Rado number for the pair of equations $$y-x=ma$$ , $$y-x=mb$$ , is the least positive integer $$R={\mathscr {R}}_d(ma,mb)$$ , if it exists, such that every 2-coloring $$\chi$$ of the integers in $$\{1,\ldots ,R\}$$ admits a solution to at least one of $$\chi (x)=\chi (x+ma)$$ , $$\chi (x)=\chi (x+mb)$$ . We show that $${\mathscr {R}}_d(ma,mb)$$ exists if and only if ab is even, and that it equals $$m(a+b-1)+1$$ in this case. We also show that there are exactly $$2^m$$ valid 2-colorings of $$[1,m(a+b-1)]$$ for the equations $$y-x=ma$$ and $$y-x=mb$$ , and use this to obtain another proof of the formula for $${\mathscr {R}}_d(ma,mb)$$ .