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,∞),+).

The first author acknowledges support recieved from the National Science Foundation via Grant DMS-0554803