On complete subsets of the cyclic group

Abstract

A subset $X$ of an abelian $G$ is said to be {\em complete} if every element of the subgroup generated by $X$ can be expressed as a nonempty sum of distinct elements from $X$. Let $A\subset \Z_n$ be such that all the elements of $A$ are coprime with $n$. Solving a conjecture of Erd\H{o}s and Heilbronn, Olson proved that $A$ is complete if $n$ is a prime and if $|A|>2\sqrt{n}.$ Recently Vu proved that there is an absolute constant $c$, such that for an arbitrary large $n$, $A$ is complete if $|A|\ge c\sqrt{n},$ and conjectured that 2 is essentially the right value of $c$. We show that $A$ is complete if $|A|> 1+2\sqrt{n-4}$, thus proving the last conjecture.