---

Claim. Every set $A$ of $n$ integers contains an admissible subset $B$, one in which two sums of distinct elements with different numbers of summands never coincide, with

$$|B|\gg\sqrt{\frac{n\log\log n}{\log n}},$$

so that $h(n)\gg(n\log\log n/\log n)^{1/2}$ in the notation of Problem 789. With Straus's upper bound $h(n)\ll n^{1/2}$ this would confine $h(n)$ to the range between $n^{1/2}(\log\log n/\log n)^{1/2}$ and a constant times $n^{1/2}$, a window of relative width $(\log n/\log\log n)^{1/2}$, and the claimant files the result on the site's proof-claim tab as a full resolution of the estimate. The route, as the claim's summary describes it: a largest admissible subset of $A$ supplies low-dimensional coordinate representations of the elements of $A$; Cramer's rule turns these into small integer images that preserve the relevant relations; the images are grouped by the exact power of a prime dividing them and by normalized residues, and the divisibility bounds so obtained are summed over a suitable interval of primes. The claim was submitted on 2026-09-12 by Samuel Korsky, the author of a preprint on progression-free subset sums for Problem 817 (source card), and declares the use of GPT Astra; its notes credit a motivating idea to two mathematicians and connect the work to that problem. The write-up, which the formal-conjectures catalog cites as S. Korsky, A near-square-root bound for an additive problem of Erdős and Straus (2026), is a document on a file-sharing service, linked above.

Submission note. Posted to erdosproblems.com as a proof claim by Samuel Korsky (account SamKorsky) on 12 September 2026, giving "GPT Astra" as the AI used:

We prove that every $n$-element set of integers contains a subset of size

$$> h(n) \gg \sqrt{\frac{n\log\log n}{\log n}}$$

whose equal subset sums have

equally many summands. Together with the classical square-root upper bound, this determines $h(n)$ up to polylogarithmic factors. The proof uses a largest admissible subset to obtain low-dimensional coordinate representations of each element in the set. Cramer’s rule then produces small integer images preserving the relevant relations. Grouping these images by prime valuations and normalized residues and summing the resulting divisibility bounds over an appropriate interval of primes gives the lower bound. Notes: Shoutout to Simone Costa and Stefano Della Fiore for the motivating idea; Simone recently built on a paper of mine (although the main idea was OpenAI's breakthrough for the distinct sumset problem) to resolve #817, and now I'm building on a paper of his to resolve #789 :).

Depends on. Straus's square-root bound, whose upper bound the completeness statement uses; the lower-bound argument itself is self-contained.

Standing. Claimed. The site's label was OPEN on 2026-09-18 and on 2026-10-06 and its commentary does not mention the claim; no arXiv version, refereed publication or independent review was found on 2026-09-18, and none of the five comments on the claim thread reviews the write-up. The site's curator, Thomas F. Bloom, comments on the thread (2026-09-12) that they keep the problem open where there is ambiguity, since nothing is gained by closing it early, and asks whether $h(n)=o(n^{1/2})$; another commenter notes that the problem's "estimate" wording asks for more than the order of growth; neither is a review, and the curator's comment is not a credit of the result. The formal-conjectures catalog has carried the lower bound since 2026-09-29 as the variant erdos_789.variants.sqrt_loglog_div_log_isBigO of ErdosProblems/789.lean, marked solved, with a formal_proof in the fork linked above at its commit of 2026-09-29; that file's Korsky section says it proves the bound of [Ko26] following its Lemma 2, works with signed relations in $A\setminus\{0\}$ rather than reducing to positive integers, uses Chebyshev's bounds in place of Mertens' theorem, and contains no sorry, axiom or native_decide; the catalog's main statement erdos_789 stays open. Being a formalization of the named claimant's result, it is a link on this page and not a claim of its own; the corpus has not built this Lean, so it gives no formalized evidence and the standing stays claimed. The lower bound in force on the problem page is Choi's $h(n)\gg(n\log n)^{1/3}$, so the claim would raise the exponent of the lower bound from $1/3$ to $1/2$ and close the gap to the upper bound up to logarithms.