If $A\subseteq\{1, ..., N\}$ with $|A| = n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then $$N \gg 2 ^ n.$$

A generalisation of the problem to sets $A \subseteq (0, N]$ of real numbers, such that the subset sums all differ by at least $1$ is proposed in [Er73] and [ErGr80].

[Er73] Erdős, P., _Problems and results on combinatorial number theory_. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.

[ErGr80] Erdős, P. and Graham, R., _Old and new problems and results in combinatorial number theory_. Monographies de L'Enseignement Mathematique (1980).