Collections of sequences having the Ramsey property only for few colours

A family C of sequences has the r-Ramsey property if for every positive integer k, there exists a least positive integer g^(r)(k) such that for every r-colouring of {1,2, . . . ,g(r)(k)} there is a monochromatic k-term member of C. For fixed integers m > 1 and 0 ≤ a < m, define a k-term a (mod m)-sequence to be an increasing sequence of positive integers {x₁, . . . , x_k} such that x_i - x_i-1 = a (mod m) for i = 2, . . . ,k. Define an m-a.p. to be an arithmetic progression where the difference between successive terms is m. Let C*_a(m) be the collection of sequences that are either a (mod m)-sequences or m-a.p.'s. Landman and Long showed that for all m ≥ 2 and 1 ≤ a < m, C*_a(m) has the 2-Ramsey property, and that the 2-Ramsey function g⁽²⁾_a(m)(k, n) , corresponding to k-term a (mod m)-sequences or n-term m-a.p.'s, has order of magnitude mkn. We show that C*_a(m) does not have the 4-Ramsey property and that, unless m/a = 2, it does not have the 3-Ramsey property. In the case where m/a = 2, we give an exact formula for g⁽³⁾_a(m)(k,n). We show that if a ≠ 0, there exist 4-colourings or 6-colourings (depending on m and a) of the positive integers which avoid 2-term monochromatic members of C*_a(m), but that there never exist such 3-colourings. We also give an exact formula for g^(r)_0(m)(k, n).