Ramsey Functions for Generalized Progressions

الأصل · EN

Given positive integers n 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 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.