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Claim. Let $h(n)$ be as in Problem 860. Then

$$h(n)\ \ge\ n\exp\!\left(\Bigl(\frac{\log2}{2}-o(1)\Bigr)\frac{\log n}{\log\log n}\right).$$

Samuel Korsky filed the claim on the site's proof-claims tab on 26 July 2026, marked as made with the AI system GPT 5.6-Pro, with a short self-contained write-up linked from the tab. The argument adapts, with small changes, a construction of Ben Green and Imre Z. Ruzsa, On the arithmetic Kakeya conjecture of Katz and Tao, Periodica Mathematica Hungarica 78 (2019), 135--151, doi:10.1007/s10998-018-0270-z; the claimant writes that the AI system noticed the connection to that paper and that the write-up stays close to it. The tab's notes add that the same improvement carries over to the lower bound of Problem 711, that it strengthens the bound of Kominers's paper Long Intervals Without Distinct Multiples of the First $n$ Positive Integers (arXiv:2607.10431, 11 July 2026), and that it answers the question at the end of that paper in the negative.

The bound then appeared in the arXiv paper Improved Bounds for Distinct Multiples in Intervals, arXiv:2607.26450, where $F(n)$ is the function of Problem 711 and $h_{\mathbb P}(n)$ is this problem's $h(n)$ less one (the site's open interval holds $h(n)-1$ integers), a shift no stated bound feels. Its first version (29 July 2026, by Kaizhe Chen alone) stated the lower bound $F(n)\ge h_{\mathbb P}(n)\ge n\exp(\frac1{50}\frac{\log n}{\log\log n})$, with constant $1/50$; its second version (13 August 2026, joint with Samuel Korsky) states Korsky's constant as Theorem 1.3, $F(n)\ge h_{\mathbb P}(n)\ge n\exp((\frac{\log2}2-o(1))\frac{\log n}{\log\log n})$. The second version's statement on AI says the authors used ChatGPT-5.6 Sol as an exploratory and proof-auditing tool. Chen's separate claim of three days later, which includes a lower bound of the same shape with an unspecified constant together with new upper bounds, has its own page.

Submission note. Posted to erdosproblems.com as a proof claim by Samuel Korsky (account SamKorsky) on 26 July 2026, giving "GPT 5.6-Pro" as the AI used:

By making small adaptions to an argument of Green and Ruzsa here: https://doi.org/10.1007/s10998-018-0270-z, one achieves the bound

$$h(n)\geq > n\exp\left( \left(\frac{\log 2}{2}-o(1)\right) \frac{\log n}{\log\log n} > \right)$$

The connection to their argument was noticed by GPT-5.6 Pro. The

attached write up is quite similar to the Green and Ruzsa paper, but because the full self-contained proof is so short I left it as is. Notes: This improvement to the lower bound naturally extends to an improvement for the lower bound Problem #711, and in particular strengthens the bound achieved by Kominers in this paper: https://scottkom.com/assets/articles/Kominers_Long_Intervals_Without_Distinct_Multiples.pdf. Furthermore, it resolves the question at the end of that paper in the negative.

Covers. The lower bound only. It improves the previously recorded lower bounds, $h(n)>(3-o(1))n$ of Erdős and Selfridge and $h(n)/n\to\infty$ of Ruzsa, to a factor $\exp(c\log n/\log\log n)$ above $n$. The upper bound and the order of magnitude of $h(n)$, which the problem asks to estimate, are not addressed; the best upper bound is on the joint page of Chen and Korsky.

Standing. The tab lists the claim as a partial proof with four comments and warns that listing is no check of correctness; the site's label is unchanged and its commentary does not record the bound. The arXiv paper has no journal reference, and no reviewer independent of the claimants has endorsed the argument. The claim stays claimed.

Depends on. Nothing on this wiki; the argument is the write-up's own.