The Ramsey property for collections of sequences not containing all arithmetic progressions

A familyB of sequences has the Ramsey property if for every positive integer k, there exists a least positive integer fB(k) such that for every 2-coloring of f1;2; : : : ; fB(k)g there is a monochromatic k-term member of B. For fixed integers m > 1 and 0 q < m, let Bq(m) be the collection of those increasing sequences of positive integers fx1; : : : ;xkg such that xi+1 xi q(mod m) for 1 i k 1. For t a fixed positive integer, denote by At the collection of these arithmetic progressions having constant difference t. Landman and Long showed that for all m 2 and 1 q < m, Bq(m) does not have the Ramsey property, whileBq(m)[Am does. We extend these results to various finite unions of Bq(m)’s andA ’s. We show that for all m 2, S q = 1m 1Bq(m) does not have the Ramsey property. We give necessary and sufficient conditions for collections of the form B [ ( S t2T At) to have the Ramsey property. We determine when collections of the form Ba(m1) [Bb(m2) have the Ramsey property. We extend this to the study of arbitrary finite unions ofBq(m)’s. In all cases considered for whichB has the Ramsey property, upper bounds are given for fB .