Ramsey properties for integer sequences with restricted gaps

A family [Figure not available: see fulltext.] of sequences has the Ramsey property if for every positive integer k, there exists a least positive integer [Figure not available: see fulltext.] such that for every 2-coloring of [Figure not available: see fulltext.] there is a monochromatic k-term member of [Figure not available: see fulltext.]. For fixed integers m > 1 and 0 ≤q < m, let [Figure not available: see fulltext.] be the collection of those increasing sequences of positive integers {x 1,..., x k } such that x i+1 ? xi ? q(mod m) for 1 ≤i ≤ k ? 1. For t a fixed positive integer, denote by [Figure not available: see fulltext.] the collection of those arithmetic progressions having constant difference t. Landman and Long showed that for all m ? 2 and 1 ≤ q < m, [Figure not available: see fulltext.] does not have the Ramsey property, while [Figure not available: see fulltext.] does. We extend these results to various finite unions of [Figure not available: see fulltext.] and [Figure not available: see fulltext.]. We show that for all m ? 2, [Figure not available: see fulltext.] does not have the Ramsey property. We give necessary and sufficient conditions for collections of the form [Figure not available: see fulltext.] to have the Ramsey property. We determine when collections of the form [Figure not available: see fulltext.] have the Ramsey property. We extend this to the study of arbitrary finite unions of [Figure not available: see fulltext.]. In all cases considered for which [Figure not available: see fulltext.] has the Ramsey property, upper bounds are given for [Figure not available: see fulltext.].