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Let $\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 < \dots <a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture byErd\H{o}s states that $a_n>c\cdot 2^n$ for some constant $c$, while the bestresult known to date is of the form $a_n>c\cdot 2^n/\sqrt{n}$. In this paper,we weaken the condition by requiring that only sums corresponding to subsets ofsize smaller than or equal to $\lambda n$ be distinct. For this case, we derivelower and upper bounds on the smallest possible value of $a_n$.

Variations on the Erd\H{o}s distinct-sums problem