Monochromatic sequences whose gaps belong to {d, 2d, ..., md}

For m and k positive integers, define a k-term h_m-progression to be a sequence of positive integers {x₁,...,_k} such that for some positive integer d, x_i+1 - x_i ∈ {d, 2d,...,md} for i = 1,..., k - 1. Let h_m(k) denote the least positive integer n such that for every 2-colouring of {1, 2,...,n} there is a monochromatic h_m-progression of length k. Thus, h₁(k) = w(k), the classical van der Waerden number. We show that, for 1 ≤ r < m, h_m(m + r) ≤ 2c(m + r - 1) + 1, where c = [m/(m - r)]. We also give a lower bound for h_m(k) that has order of magnitude 2k²/m. A precise formula for h_m(k) is obtained for all m and k such that k ≤ 3m/2.