New Proofs for the Disjunctive Rado Number of the Equations x1 - x2 = a and x1 - x2 = 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 = R-d (ma, mb), if it exists, such that every 2-coloring chi of the integers in {1, ..., R} admits a solution to at least one of chi(x) = chi(x + ma), chi(x) = chi(x + mb). We show that 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 R-d (ma, mb).