Sum-Free Subsets of Right Cancellative Semigroups

For a natural number k ≥ 2 letρ = ρ(k) be the smallest natural number which does not dividek − 1. We show that for any subset A of a right cancellative semigroup S which contains no solutions of the equation x1 + ⋯ + xk = y there is an element s inS such that the setsA, A + s, . . . ,A + (ρ − 1)sare pairwise disjoint. In particular, if S is finite, such a set A has at most | S | / ρ elements. This estimate is sharp.