On the degree of regularity of generalized van der Waerden triples

Details On the degree of regularity of generalized van der Waerden triples On the degree of regularity of generalized van der Waerden triples

2005-07-28

arxiv.org/abs/math/0507588v1

Mathematics

Combinatorics

5 pages

Scientific paper

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 $(a,b)$.

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