Sum-full sets are not zero-sum-free (Q2032257)

A subset \(A\) of an abelian group is called \textit{sum-full} if every element of \(A\) is a sum of two other elements, possibly equal to each other. The subset is \textit{zero-sum} if the sum of its elements is equal to \(0\); it is \textit{zero-sum-free} if it does not contain itself a non-empty zero-sum subset. In 2010, Gjergji Zaimi posed the following problem at the MathOverflow web: Can a finite, nonempty, sum-full set of real numbers be zero-sum-free? In this paper, the authors present a complete solution to this problem. \textbf{Theorem 1.} Let \(A\) be a finite, nonempty subset of an abelian group. If \(A\) is sum-full, then it is not zero-sum-free; that is, if every element of A is representable as a sum of two other elements, then A has a nonempty zero-sum subset.