On the Size of Dissociated Bases

@article{Lev2011OnTS,
  title={On the Size of Dissociated Bases},
  author={Vsevolod F. Lev and Raphael Yuster},
  journal={Electron. J. Comb.},
  year={2011},
  volume={18}
}
We prove that the sizes of the maximal dissociated subsets of a given finite subset of an abelian group differ by a logarithmic factor at most. On the other hand, we show that the set $\{0,1\}^n\subseteq\mathbb{Z}^n$ possesses a dissociated subset of size $\Omega(n\log n)$; since the standard basis of $\mathbb{Z}^n$ is a maximal dissociated subset of $\{0,1\}^n$ of size $n$, the result just mentioned is essentially sharp. 
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A set S of positive integers has distinct subset sums if the set {∑ x∈X x : X ⊂ S } has 2|S| distinct elements. Let f(n) = min{maxS : |S| = n and S has distinct subset sums}. In 1931 Paul Erdős
the electronic journal of combinatorics
  • the electronic journal of combinatorics
  • 2011