Avoiding Monochromatic Sequences With Special Gaps

Original · EN

For S a set of positive integers, and k and r fixed positive integers, denote by f(S,k;r) the least positive integer n (if it exists) such that within every r-coloring of {1,2,...,n} there must be a monochromatic sequence {x₁,x₂,...,xₖ} with xᵢ-xᵢ₋₁ ∈ S for 2 ≤ i ≤ k. We consider the existence of f(S,k;r) for various choices of S, as well as upper and lower bounds on this function. In particular, we show that this function exists for all k if S is an odd translate of the set of primes and r=2.