A finite set of naturals $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A\subseteq{1, ..., N}$ and the sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$
A finite set of real numbers is said to be sum-distinct if all the subset sums differ by at least $1$.
If $A\subseteq {1,\ldots,N}$ with $\lvert A\rvert=n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then $N \gg 2^{n}.$
Erdős called this 'perhaps my first serious problem'. The powers of $2$ show that $2^n$ would be best possible here. The trivial lower bound is $N \gg 2^{n}/n$, since all $2^n$ distinct subset sums must lie in $[0,Nn)$. Erdős and Moser [Er56] proved $ N\geq (\tfrac{1}{4}-o(1))\frac{2^n}{\sqrt{n}}.$ A number of improvements of the constant have been given (see [St23] for a history), with the current record $\sqrt{2/\pi}$ first proved in unpublished work of Elkies and Gleason. Two proofs achieving this constant are provided by Dubroff, Fox, and Xu [DFX21], who in fact prove the exact bound $N\geq \binom{n}{\lfloor n/2\rfloor}$. In [Er73] and [ErGr80] the generalisation where $A\subseteq (0,N]$ is a set of real numbers such that the subset sums all differ by at least $1$ is proposed, with the same conjectured bound. (The second proof of [DFX21] applies also to this generalisation.) This problem appears in Erdős' book with Spencer [ErSp74] in the final chapter titled 'The kitchen sink'. As Ruzsa writes in [Ru99] "it is a rich kitchen where such things go to the sink". See also [350].
The trivial lower bound is $N \gg 2^n / n$.
Erdős and Moser [Er56] proved $$ N \geq (\tfrac{1}{4} - o(1)) \frac{2^n}{\sqrt{n}}. $$
[Er56] Erdős, P., Problems and results in additive number theory. Colloque sur la Th'{E}orie des Nombres, Bruxelles, 1955 (1956), 127-137.
A number of improvements of the constant $\frac{1}{4}$ have been given, with the current record $\sqrt{2 / \pi}$ first provied in unpublished work of Elkies and Gleason.
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).
The minimal value of $N$ such that there exists a sum-distinct set with three elements is $4$.
The minimal value of $N$ such that there exists a sum-distinct set with five elements is $13$.
The minimal value of $N$ such that there exists a sum-distinct set with nine elements is $161$.