On the degree of regularity of generalized van der Waerden triples

Abstract Let 1 ⩽ a ⩽ b be integers. A triple of the form ( x , a x + d , b x + 2 d ) , 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 ) .