A problem of Erdős and Sárkőzy. The greedy algorithm shows that \[\lvert A\rvert\geq (1-o(1))\log_3 n\] is possible, but Erdős and Sárkőzy speculated that $\lvert A\rvert=(1-o(1))\log_2n$ is possible.

In [Er98] Erdős reports (but gives no reference) that Sándor has proved that $\lvert A\rvert=(1-o(1))\log_2 n$ is achievable, taking $A=\{ 2^i+m2^m : 0\leq i<m\}$ and $n=2^{m-1}+m2^m$.

Erdős, Lev, Rauzy, Sándor, and Sárközy [ELRSS99] proved that \[\lvert A\rvert > \log_2 n -1\] is achievable, taking $A=\{2^m-2^{m-1},2^m-2^{m-2},\ldots,2^m-1\}$. This property also implies that $\sum_{a\in S}a$ are distinct for distinct subsets $S$, whence [1] implies \[\lvert A\rvert \leq \log_2 n+\tfrac{1}{2}\log_2\log n+O(1),\] and likely $\lvert A\rvert\leq \log_2n+O(1)$.

See also [13].