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

A family ℬ of sequences has the Ramsey property if for every positive integer k, there exists a least positive integer f_ℬ(k) such that for every 2-coloring of {1, 2, . . . , f_ℬ(k)} there is a monochromatic k-term member of ℬ. For fixed integers m > 1 and 0 ≤ q < m, let ℬ_q(m) be the collection of those increasing sequences of positive integers {x₁ , . . . , x_k} such that x_i+1 - x_i = q(mod m) for 1 ≤ i ≤ k - 1. For t a fixed positive integer, denote by script A sign_t the collection of those arithmetic progressions having constant difference t. Landman and Long showed that for all m ≥ 2 and 1 ≤ q < m, ℬ_q(m) does not have the Ramsey property, while ℬ_q(m) ∪ script A sign_m does. We extend these results to various finite unions of ℬ_q(m)'s and script A sign_t's. We show that for all m ≥ 2, ∪_q=1^m-1 ℬ_q(m) does not have the Ramsey property. We give necessary and sufficient conditions for collections of the form ℬ_q(m) ∪ (∪_{t ∈ T} script A sign_t) to have the Ramsey property. We determine when collections of the form ℬ_a(m1) ∪ ℬ_b(m2) have the Ramsey property. We extend this to the study of arbitrary finite unions of ℬ_q(m)'s. In all cases considered for which ℬ has the Ramsey property, upper bounds are given for f_ℬ.