Let M be a given nonempty set of positive integers and S any set of nonnegative integers. Let δ(S) denote the upper asymptotic density of S. We consider the problem of finding µ(M) := sup S δ(S), where the supremum is taken over all sets S satisfying that for each a, b ∈ S, a − b ∉ M. In this paper we discuss the values and bounds of µ(M) where M = {a, b, a + nb} for all even integers and for all sufficiently large odd integers n with a < b and gcd(a, b) = 1.