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Claim. Let $R(N)$ be the largest size of a set $A\subseteq\{1,\ldots,N\}$ whose subset sums $\sum_{n\in S}1/n$, $S\subseteq A$, are pairwise distinct, and let $\log_j$ be the $j$-fold iterated logarithm. Then

$$R(N)\ \asymp\ \log S(N)\ \asymp\ \frac{N}{\log N}\prod_{j=3}^{k}\log_jN,$$

where $S(N)$ counts the distinct reciprocal subset sums of $\{1,\ldots,N\}$ and $k=k(N)$ is an index with $1\le\log_kN=O(1)$, such as the last index with $\log_kN\ge1$; every such choice gives the same order. The claim's summary writes $\kappa(N)$ without defining it; the audit note in the claimants' Lean repository at the pinned commit takes $\log_{\kappa(N)}N>2$ and says any fixed threshold above $1$ gives the same order. The upper bound follows from $2^{R(N)}\le S(N)$, since the $2^{R(N)}$ subset sums of an extremal set are distinct values among those $S(N)$ counts, together with the claimants' upper bound for $\log S(N)$; the lower bound is the set $\mathcal U(N)$ from the proof of the refereed Theorem 1 of Bettin, Grenié, Molteni and Sanna, which has distinct subset reciprocal sums and the stated size. The claim determines the order of magnitude of $R(N)$, the question of Problem 321, and no asymptotic formula, as its notes say; the site reads the order of magnitude as the resolution, and this page follows that reading. It answers the monograph's rider question, whether $\log S(N)/R(N)\to\infty$, in the negative.

Submission note. Posted to erdosproblems.com as a proof claim by RayYoung, Keheng Zhu, Yanping Luo (account RayYoung) on 15 July 2026, giving "GPT 5.6 Sol Pro" as the AI used, which the site marks as accepted as correct:

We prove that the largest reciprocal-dissociated subset of $\{1,\ldots,N\}$ has order

$$\frac{N}{\log N}\prod_{j=3}^{\kappa(N)}\log_jN.$$

The upper bound

follows from $2^{R(N)}\le S(N)$, which is a natural development based on Erdos' ideas, and the lower bound from the dissociated set constructed in the work of Bettin, Grenié, Molteni, and Sanna. Notes: The manuscript submitted for Problem #320 also contains a proposed order-of-magnitude resolution of this problem, although it does not yet provide an exact asymptotic formula. The method may admit further refinement, possibly leading to the determination of a leading asymptotic constant. We warmly welcome comments, corrections, and further discussion from the community.

Depends on. The accepted upper bound for log S(N) supplies the upper half of the claim through $2^{R(N)}\le S(N)$; without it the refereed bounds determine $R(N)$ only up to an unbounded factor $\log_rN$. The dissociated set of Bettin, Grenié, Molteni and Sanna supplies the lower half.

Provenance. The claim was submitted to the site's proof-claim tab on 15 July 2026 by the account RayYoung for RayYoung, Keheng Zhu and Yanping Luo, hours after the same authors' claim on Problem 320; its notes say that the manuscript submitted for Problem 320 also contains a proposed order-of-magnitude resolution of this problem without an exact asymptotic, and the claim's tab names the AI system GPT 5.6 Sol Pro. The manuscript sits behind the same Overleaf read link, which served no document to a request on 2026-09-18; the account of the upper-bound argument is on the Problem 320 claim page. The Lean repository at the pinned commit of 11 July 2026 proves the finite dissociation bridge for this problem, that $\mathcal U(N)$ is dissociated and that $2^{|\mathcal U(N)|}$ is at most the number of subset sums (BGMSU_dissociated, pow_card_BGMSU_le_harmonic_subsetSums), and nothing about the order of $R(N)$, and it has not been built by this corpus, so it is a link and no formalized evidence.

Acceptance. The site's curator, Thomas Bloom, marked the claim on the tab as accepted by the site as correct and marked the problem resolved on 16 July 2026 with a thread comment saying that the order of magnitude of $R(N)$ is now known and that finer questions such as an asymptotic remain open; the problem page's commentary attributes the upper bound to the AI system prompted by the claimants and points to Problem 320. The curator is independent of the claimants, and that acceptance is the reviewed evidence. No refereed publication and no independent review were found on 2026-09-18. The one comment under the claim (2026-10-07) is the curator's, of 16 July 2026: it says that the proof is correct, that the lower bound is from the earlier work the claim cites, and that the upper bound follows at once from the resolution of Problem 320.