On the degree of regularity of generalized van der Waerden triples

AbstractLet 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)