  Let $\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 < \dots <
a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture by
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 weaken the condition by requiring that only sums corresponding to subsets of
size smaller than or equal to $\lambda n$ be distinct. For this case, we derive
lower and upper bounds on the smallest possible value of $a_n$.
