Avoiding monochromatic sequences with gaps in a fixed translation of the primes - Landman, Bruce M.; Ventullo, Kevin (Utilitas Mathematica, 2010)
For a set D of positive integers, a sequence a1 < a 2 < ⋯ ak is called an k-term D-diffsequence if ai - ai-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.