  Let $1 \leq a \leq b$ be integers. A triple of the form $(x,ax+d,bx+2d)$,
where $x,d$ are positive integers is called an {\em (a,b)-triple}. The {\em
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 $\mathbb{N}$ admits
a monochromatic $(a,b)$-triple. We settle, in the affirmative, the conjecture
that dor$(a,b) < \infty$ for all $(a,b) \neq (1,1)$. We also disprove the
conjecture that dor($a,b) \in \{1,2,\infty\}$ for all $(a,b)$.
