A construction for sets of integers with distinct subset sums (Q1379125)

Summary: A set \(S\) of positive integers has distinct subset sums if there are \(2^{|S|}\) distinct elements of the set \(\{ \sum_{x \in X} x: X \subset S\}\). Let \((f(n) = \min\{ \max S: |S|=n \) and \(S\) has distinct subset sums\}. Erdős conjectured \(f(n) \geq c2^{n}\) for some constant \(c\). We give a construction that yields \(f(n) < 0.22002 \cdot 2^{n}\) for \(n\) sufficiently large. This now stands as the best known upper bound on \(f(n)\).