  Let $\Sigma=\{a_1, \ldots , a_n\}$ be a set of positive integers with $a_1 <
\ldots < a_n$ such that all $2^n$ subset sums are pairwise distinct. A famous
conjecture of Erd\H{o}s states that $a_n>C\cdot 2^n$ for some constant $C$,
while the best result known to date is of the form $a_n>C\cdot 2^n/\sqrt{n}$.
In this paper, we propose a generalization of the Erd\H{o}s distinct sum
problem that is in the same spirit as those of the Davenport and the
Erd\H{o}s-Ginzburg-Ziv constants recently introduced in \cite{CGS} and in
\cite{CS}. More precisely, we require that the non-zero evaluations of the
$m$-th degree symmetric polynomial are all distinct over the subsequences of
$\Sigma$ whose size is at most $\lambda n$, for a given $\lambda\in (0,1]$,
considering $\Sigma$ as a sequence in $\mathbb{Z}^k$ with each coordinate of
each $a_i$ in $[0,M]$. If $\mathcal{F}_{\lambda,n}$ denotes the family of
subsets of $[1,n]$ whose size is at most $\lambda n$, our main result is that,
for each $k,m,$ and $\lambda$, there exists an explicit constant
$C_{k,m,\lambda}$ such that $$ M\geq C_{k,m,\lambda} \frac{(1+o(1))
|\mathcal{F}_{\lambda,n}|^{\frac{1}{mk}}}{n^{1 - \frac{1}{2m}}}.$$
