A famous open conjecture of Paul Erdos, first stated about 80 years ago, is that if all subset sums of an integer set $S\subset[1,n]$ are pairwise distinct, then $|S|<\log_2n+O(1)$ as $n\to\infty$. (Here $\log_2$ denotes the base-$2$ logarithm.) In modern terms, a subset of an abelian group, all of whose subset sums are pairwise distinct, is called *dissociated*. Similarly to Erdos' original problem, one can ask how large can dissociated subsets of other "natural" sets in abelian groups be; say, you are asking what is the largest possible size of a dissociated subset of the set $\{0,1\}^n\subset{\mathbb R}^n$.

Problems of this sort were studied in a joint paper ["On the size of dissociated bases"][1] by Raphy Yuster and myself. We have shown that the sizes of any two maximal dissociated subsets of a given finite subset of an abelian group differ by a logarithmic factor at most. Addressing the specific set $\{0,1\}^n$, we proved that, denoting by $M_n$ the largest size of its dissociated subset, we have
  $$ (1+o(1))n\log_2 n/\log_2 9 < M_n < (1+o(1))n\log_2 n. $$
It is an open problem to determined the best possible coefficients.

[1]: http://math.haifa.ac.il/~seva/Papers/DisBases.pdf