Abstract. Let α be the supremum of all δ such that there is a sequence ⟨A_{n}⟩_{n=1}^{∞} of measurable subsets of (0, 1) with the property that each A_{n} has measure at least δ and for all n, m ∈ ℕ, A_{n}∩A_{m}∩ A_{n+m} = ∅. For k ∈ ℕ, let α_{k} be the corresponding supremum for finite sequences ⟨A_{n}⟩_{n=1}^{k}. We show that α = lim_{k→∞} α_{k} and find the exact value of α_{k} for k ≤ 41. In the process of finding these exact values, we also determine exactly the number of maximal sum free subsets of {1, 2, …, k} for k ≤ 41. We also investigate the size of sets ⟨A_{x}⟩_{x∈S} with A_{x} ∩ A_{y} ∩ A_{x+y} = ∅ where S is a subsemigroup of ((0, ∞), +).