---

Claim. Every set of $n$ integers has a sum-free subset of at least

$$\frac n3+(\log n)^{1/3-o(1)}$$

elements: $f(n)\ge n/3+(\log n)^{1/3-o(1)}$ in the notation of Problem 792, a quantitative improvement on the $c\log\log n$ of the claimant's preprint. The route, as the claim's summary describes it, is an approach the claimant had considered before the one used in the preprint and worked out in a conversation with the model GPT 6 Astra, as the tab names it: the preprint's dense model and its partial sifting results are combined with structural results on functions of small Fourier algebra norm, in the manner of Sanders's papers. The claim was submitted on 2026-09-09 by Benjamin Bedert, the author of the preprint, under the account bkbedert, and listed by the site with no kind; its note says that the write-up is AI-generated and that a human-readable version is intended. The write-up is a document on a file-sharing service, linked above, and is not examined in this corpus.

Submission note. Posted to erdosproblems.com as a proof claim by Benjamin Bedert (account bkbedert) on 9 September 2026, giving "GPT 6 Astra" as the AI used:

In a conversation with GPT 6 astra, we got an earlier approach (that I considered before the one used in my current arXiv preprint) to work. This yields a somewhat better quantitative bound $f(n)\geqslant \frac n3+(\log n)^{1/3-o(1)}$. Some ideas behind this improvement are to combine the dense model and partial sifting results from my arXiv preprint with some structural results for functions with small Fourier algebra norm as in the papers of Sanders. Notes: The current write-up is AI-generated (due to time constraints), but I intend to produce a human-readable version in the near future.

Covers. A lower bound for the second-order term, $f(n)-n/3\ge(\log n)^{1/3-o(1)}$. Not covered: the upper bound, which is Eberhard, Green and Manners's $o(n)$, and the true order of the second-order term.

Depends on. Bedert's preprint, whose dense model and partial sifting results the argument reuses.

Standing. Claimed. The site's label was OPEN on 2026-09-18 and its commentary adopts the preprint's $c\log\log n$, not this bound; no arXiv version, refereed publication or independent review was found on that date, and the two comments on the claim thread, a congratulation and a question about the write-up's contents, review nothing. The write-up is not examined in this corpus.