On the determination of sets by sets of sums of a certain order

1. Introduction * Let X = {xl9 •••, xn} be a set of (not necessarily distinct)1 elements of a torsion free Abelian group. Define PS(X) =!%h + %i2 + + xig I ii < it < < i.}. Thus PS(X) has (£} (not neces-sarily distinct) elements. We introduce the equivalence relation X ~ Y if and only if P8(X) = P8(Y). Let Fs(n) be the greatest number of set