On the degree of regularity of generalized van der Waerden triples

Original · EN

Let 1 ≤ a ≤ b be integers. A triple of the form (x,ax+d,bx+2d), where x,d are positive integers is called an (a,b)-triple. The degree of regularity of the family of all (a,b)-triples, denoted dor(a,b), is the maximum integer r such that every r-coloring of N admits a monochromatic (a,b)-triple. We settle, in the affirmative, the conjecture that dor(a,b) < ∞ for all (a,b) ≠ (1,1). We also disprove the conjecture that dor(a,b) ∈ {1,2,∞} for all (a,b).