  For a finite abelian group $G$ written additively, and a non-empty subset
$A\subset [1,\exp(G)-1]$ the weighted Davenport Constant of $G$ with respect to
the set $A$, denoted $D_A(G)$, is the least positive integer $k$ for which the
following holds: Given an arbitrary $G$-sequence $(x_1,\ldots,x_k)$, there
exists a non-empty subsequence $(x_{i_1},\ldots,x_{i_t})$ along with $a_{j}\in
A$ such that $\sum_{j=1}^t a_jx_{i_j}=0$. In this paper, we pose and study a
natural new extremal problem that arises from the study of $D_A(G)$: For an
integer $k\ge 2$, determine $\fD_G(k):=\min\{|A|: D_A(G)\le k\}$ (if the
problem posed makes sense). It turns out that for $k$ `not-too-small', this is
a well-posed problem and one of the most interesting cases occurs for $G=\Z_p$,
the cyclic group of prime order, for which we obtain near optimal bounds for
all $k$ (for sufficiently large primes $p$), and asymptotically tight (up to
constants) bounds for $k=2,4$.
