On the degree of regularity of generalized van der Waerden triples
Abstract
Let
1≤a≤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
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)
.