Avoiding monochromatic sequences with gaps in a fixed translation of the primes

Abstract

For a set D of positive integers, a sequence {a₁ < a ₂ < ⋯ a_k} is called an k-term D-diffsequence if a_i - a_i-1 ∈ D for all i ∈ {2,..., k}. For a positive integer r, a set of positive integers D is r-accessible if every r-coloring of ℤ⁺ has arbitrarily long monochromatic D-diffsequences. The largest r such that D is r-accessible is called the degree of accessibility of D. It is already known that each odd translation of the set of primes, P + t, is 2-accessible. We offer new results on the accessibility of translations the primes. The main result is that for any c ≥ 2, the degree of accessibility of P + c does not exceed the smallest prime factor of c.