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Claim. Let $f(N)$ be the extremal function of Problem 301. The manuscript A positive-density improvement for all-length unit-fraction-free sets (Theorem 1.1, p. 1) states that there is an absolute $\varepsilon>0$ such that $f(N)\ge(1/2+\varepsilon)N$ for all sufficiently large $N$. If correct, this answers the problem's particular question, whether $f(N)=(1/2+o(1))N$, in the negative; the monograph's question of 1980, whether the interval example $(N/2,N]$ can be beaten by a positive proportion, would be answered yes. The claim's own account of the route: a set inside $(N/3,N]$ admits no relation with three or more terms, since a $k$-term relation forces $ak<N$; the construction keeps a centered-regular subset of the top half $[\lceil N/2\rceil,N]$ and adjoins the centered-regular odd integers of $(N/3,N/2)$ free of prime factors below a parameter $L$, so that any surviving two-term relation $1/a=1/b+1/c$ has both $b$ and $c$ even and can be written $a=trs$, $b=tr(r+s)$, $c=ts(r+s)$; Theorem 3.1 of de la Bretèche and Tenenbaum, a mean-value bound for arithmetic functions at the three linear forms $r$, $s$, $r+s$, together with Tenenbaum's one-variable mean-value theorem for multiplicative functions, show that the elements involved in such relations number far fewer than the $\asymp N\rho_L$ adjoined ones, $\rho_L=\prod_{p<L}(1-1/p)$, and deleting them leaves a relation-free set of size at least $N/2+N\rho_L/24$.

Submission note. Posted to erdosproblems.com as a proof claim by Donald Della Pietra (account dondellapietra) on 30 July 2026, giving "GPT 5.6 Sol" as the AI used:

There is an absolute $\varepsilon>0$ with $f(N)\geq(1/2+\varepsilon)N$ for all large $N$, where $f(N)$ is the largest $A\subseteq{1,\ldots,N}$ with no $1/a=1/b_1+\cdots+1/b_k$ for distinct $a,b_1,\ldots,b_k\in A$ and any $k\geq2$; this disproves the Erdős–Graham guess $f(N)=(1/2+o(1))N$. Since $ak<N$, a set supported in $(N/3,N]$ admits only $k=2$ relations, and making the elements added in $(N/3,N/2)$ $L$-rough (hence odd) forces both endpoints even, giving coordinates $a=trs$, $b=tr(r+s)$, $c=ts(r+s)$. Imposing centered prime-factor regularity, a one-variable mean-value estimate together with a three-linear-form theorem bounds the conflicting heads well below the added source $\asymp N\rho_L$, $\rho_L=\prod_{p<L}(1-1/p)$, so deleting them leaves $|A|\geq N/2+N\rho_L/24$. Notes: The formalization has no project-local axioms; it does not formalize Tenenbaum III.3.5 or de la Bretèche–Tenenbaum Thm 3.1 as stated, but proves explicit surrogates sufficient here, so the Lean and the paper are independent routes to the same theorem. The pinned Mathlib is an unmerged Mertens fork branch. Developed using GPT 5.6 Sol.

Covers. The particular question, answered in the negative, and a lower bound $f(N)\ge(1/2+\varepsilon)N$ with an unspecified absolute $\varepsilon$. Not covered: the estimate of $f(N)$ beyond that, in particular the asymptotic constant.

Standing. Claimed. The claimant is the human submitter, Donald Della Pietra, who filed the claim as partial on the site's proof-claim tab on 30 July 2026 naming the system GPT 5.6 Sol; the manuscript's Section 8 (p. 9) says that AI systems gave substantial assistance and that the mathematical claim is unrefereed. The manuscript is the ten-page PDF in the claimant's repository, linked above at the repository's head commit of 30 July 2026; it has no arXiv version, no journal record and no independent review. The same repository holds a Lean 4 development that the claimant describes as proving the lower bound end to end with no sorry and no axioms beyond propext, Classical.choice and Quot.sound, against a pinned fork of Mathlib carrying an unmerged file on Mertens' theorem; the claim's notes say that the two cited analytic inputs are not formalized as stated but replaced by explicit surrogates proved in Lean, so that the paper and the Lean are presented as independent routes. The proof was not checked and nothing was built, so the repository gives no formalized evidence. The repository's head commit also announces a companion upper bound, $f(N)\le(15437/19344+o(1))N\approx0.798N$, with a divisor-block certificate at $M=60$ kernel-checked and the full optimization at $M=5040$ not formalized; that result has its own claim page. Two comments stand under the claim: one reports that a GPT review found no fatal issue but asked for Proposition 4.1 to be expanded, and one remarks that Sawin's recent framework for Problem 327 is what makes the argument possible; neither is an acceptance. The site's label is OPEN (page last edited 16 January 2026; as of 2026-10-07), and the tab's standing notice says that listing a claim implies no examination. The lower bound is the input of Khanukov's claim on Problem 302.