On the degree of regularity of generalized van der Waerden triples

Abstract

Let $1\le a\le b$ 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 $(a,b)\ne (1,1)$. We also disprove the conjecture that dor($a,b)\in \{1,2,\infty \}$ for all $(a,b)$.

5 pages