Smallest possible maximum in sum-distinct integer sets
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An n-element set of positive integers is sum-distinct if all its subset sums differ. Must its largest element be at least c·2ⁿ for one universal c>0, or can the ratio to 2ⁿ become arbitrarily small?
Why it matters and what counts as progress
Why it matters: The question asks for the correct scale between known exponential constructions and lower bounds of order 2ⁿ/√n.
What counts as progress: A uniform positive lower constant, or constructions with maximum at most ε·2ⁿ for arbitrarily large n for every ε>0; effective size dependence is an additional target.
How the problem developed
- 1931 · Open problem [1]
Paul Erdős Posed the problem of minimizing the largest element of an n-element positive-integer set with distinct subset sums, conjecturing a lower bound proportional to 2^n.
- 1955 · Lower bound [2]
Paul Erdős and Leo Moser Established a lower bound of order 2^n/√n for the maximum element required by an n-element sum-distinct set.
- 2020 · Lower bound [3]
Quentin Dubroff, Jacob Fox and Max Wenqiang Xu Improved the asymptotic lower bound to the central-binomial scale, giving the best-known lower-bound order before the later reported disproof.
- 2026 · Counterexample claim [4]
Tom Adamczewski and GPT-6 Astra The project repository reports a Lean-formalized disproof of the conjectured positive-constant lower bound by constructing sum-distinct sets with maximum o(2^n).
AI contribution
GPT-6 Astra disproves Erdős’s sum-distinct lower-bound conjecture
For every ε>0, arbitrarily large n admit an n-element sum-distinct set of positive integers with maximum element N≤ε·2^n. The formal statement excludes N=0 and gives no effective n as a function of ε.
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GPT-6 Astra disproves Erdős’s sum-distinct lower-bound conjecture