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Claim. For increasing sequences of positive integers in which no term is a sum of two or more distinct earlier terms, the sum-free sequences of Problem 876, the write-up claims two theorems, rendered here from the claim's summary. First, let $F\ge1$ be nondecreasing and slowly varying; a sum-free sequence whose gaps $a_{n+1}-a_n$ are of the order $nF(n)$ (the summary writes the gap size as $\Omega(nF(n))$) exists if and only if

$$\int^{\infty}\frac{dx}{xF(x)}<\infty .$$

Second, a sum-free sequence can have gaps $O(n^c)$ and at the same time infinitely many bounded gaps when $c=(1+\sqrt{13})/2$ and for no smaller $c$. What follows in this paragraph is inferred on this page and is not a statement of the write-up. Since a sparse sequence such as the powers of $2$ is sum-free with gaps as large as one likes, the first theorem can only concern how small the gaps can be kept, a sequence all of whose gaps are at most a constant multiple of $nF(n)$; so read, $F=1$ or $F=\log$ makes the integral diverge, so gaps of order $n$ or $n\log n$ throughout are excluded, while $F=(\log x)^{1+\varepsilon}$ makes it converge, so gaps of order $n(\log n)^{1+\varepsilon}$ are attainable for every $\varepsilon>0$, and the theorem answers the first question with a threshold at the convergence of the integral and the second question no. On the methods, the summary says that the negative half rests on the completeness theorem of Bergelson and Simmons, a reduction on divisibility and estimates from continued fractions; that the sequences showing the integral criterion sharp consist of integers whose fractional parts, after a fixed dilation, fall into shrinking intervals; and that the bounded-gap theorem uses progressions inside the set of subset sums together with a construction by congruences. The claim was submitted to the site's proof-claims tab on 2026-09-22 by Samuel Korsky, who credits GPT Astra and files it as a full resolution, writing in the claim notes that the gap question admits several readings and that the two theorems answer it in a way Korsky considers complete, and that the write-up would still be revised. The write-up is a document on a file-sharing service, linked above (read status: unread); only the claim's summary and notes were checked. Korsky had earlier posted, on the problem's discussion thread on 23 June 2026, a sketch of a log-free form of Łuczak and Schoen's density bound, recorded on the problem page.

Submission note. Posted to erdosproblems.com as a proof claim by Samuel Korsky (account SamKorsky) on 22 September 2026, giving "GPT Astra" as the AI used:

For increasing sequences in which no term is a sum of two or more distinct earlier terms, the paper proves: (1) Gaps $\Omega(nF(n))$ are possible exactly when $\int^\infty dx/(xF(x))$ converges, for nondecreasing slowly varying $F\ge1$. (2) The smallest exponent permitting gaps $O(n^c)$ together with infinitely many bounded gaps is $c=(1+\sqrt{13})/2$. The nonexistence proof combines the Bergelson–Simmons completeness theorem with a divisibility reduction and continued-fraction estimates. The integral-bound constructions select integers whose fractional parts lie in shrinking intervals. The bounded-gap result uses arithmetic progressions of subset sums and an explicit construction using congruences. Notes: Erdős's question of how small the gaps can be has many interpretations, but I think the two results proved answer it in a way that can be considered a "full" resolution of the problem. I have cleaned up the write-up somewhat, but will be continuing to improve it over the next week or two.

Depends on. No page of this wiki. The completeness theorem of Bergelson and Simmons and the subset-sum results the summary names are literature results, not pages here. The partial claim of Price, that $\limsup(a_{n+1}-a_n)/n=\infty$, would be contained in the first theorem's impossibility half under the reading drawn above, an inference made on this page; the summary does not cite it as an input.

Standing. Claimed. The site's label is OPEN and its commentary does not mention the claim (page without a last-edited date, as of 2026-10-06); the claim thread had no comments, and the curator records no acceptance. No arXiv version or journal record of the write-up was found on 2026-10-07, and there is no independent review. The claim is filed as full, so the problem's derived standing is claimed; the reciprocal-sum question of the site's commentary is not addressed by it.