On a combinatorial problem of Erdős, Kleitman and Lemke

In this paper, we study a combinatorial problem originating in the following conjecture of Erdős and Lemke: given any sequence of nn divisors of nn, repetitions being allowed, there exists a subsequence the elements of which are summing to nn. This conjecture was proved by Kleitman and Lemke, who then extended the original question to a problem on a zero-sum invariant in the framework of finite Abelian groups. Building among others on earlier works by Alon and Dubiner and by the author, our main theorem gives a new upper bound for this invariant in the general case, and provides its right order of magnitude.