On the Degree of Regularity of Generalized van der Waerden Triples

Let a and b be positive integers such that a ≤ b and (a, b) \not= (1, 1). We prove that there exists a 6-coloring of the positive integers that does not contain a monochromatic (a, b)-triple, that is, a triple (x, y, z) of positive integers such that y = ax+d and z = bx+2d for some positive integer d. This confirms a conjecture of Landman and Robertson.