The Two-Colour Rado Number for the Equation ax + by = (a + b)z

Abstract

For relatively prime positive integers a and b, let n = R(a,b) denote the least positive integer such that every 2-colouring of [1, n] admits a monochromatic solution to ax + by = (a + b)z with x, y, z distinct integers. It is known that R(a,b) ≤ 4(a+b) +1. We show that R(a,b) = 4(a+b) +1, except when (a, b) = (3, 4) or (a, b) = (1, 4k) for some k≥1, and R(a,b) = 4(a+b) − 1 in these exceptional cases.