Ramsey Functions for Generalized Progressions

Given positive integers m and k, a k-term semi-progression of scope m is a sequence x1, x2, ..., xk such that xj+1 − xj ∈ {d, 2d, . . . ,md}, 1 ≤ j ≤ k − 1, for some positive integer d. Thus an arithmetic progression is a semi-progression of scope 1. Let Sm(k) denote the least integer for which every 2-coloring of {1, 2, ..., Sm(k)} yields a monochromatic k-term semi-progression of scope m. We obtain an exponential lower bound on Sm(k) for all m = O(1). Our approach also yields a marginal improvement on the best known lower bound for the analogous Ramsey function for quasi-progressions, which are sequences whose successive differences lie in a small interval.