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Claim. Let $f(N)$ be the largest size of a subset of $\{1,\ldots,N\}$ with no three distinct elements whose three pairwise least common multiples are equal. Then $f(N)=o(N)$ as $N\to\infty$: every set of positive upper density contains such a triple. This is Theorem 1.1 of S. Wang, A Proposed Complete Solution to Erdős Problem 536 (33 pages, hosted in the author's GitHub repository; the file was last changed on 22 July 2026 and is linked above at the repository's commit of 2 August 2026), and it answers yes the question "is it true that $f(N)=o(N)$?" of Problem 536. The corpus's card for the manuscript is wang_2026_proposed_complete_solution_erdos_problem_536, with the statement, the pair-product lemma and the manuscript's own outline on its result page.

Submission note. Posted to erdosproblems.com as a proof claim by Shouqiao Wang (account ShouqiaoWang) on 14 July 2026, giving "GPT-5.6 Sol" as the AI used:

We prove that $f(N)=o(N)$. Every forbidden triple can be written as $(txy,\,txz,\,tyz)$, with $x,y,z$ pairwise coprime. For squarefree integers, this becomes three prime supports with the same pairwise unions. We arrange these supports in an $\mathbb{F}_3$-cube: each coordinate contains three disjoint prime blocks, and a vertex chooses two of them. Every affine line then gives a forbidden triple, so the cap-set bound gives an exponential saving. The cutoff $(n\le N)$ means that the cube vertices must have almost equal products while still looking like ordinary random prime supports. Separated prime bands make this possible. A first-moment estimate produces enough balanced choices; a two-pivot second-moment estimate controls the bias introduced by conditioning on balance; and randomizing over many spare bands removes the remaining bias. The resulting cubes carry the cap-set saving through the cutoff, giving $f(N)=o(N)$.

Covers. The density question only: $f(N)=o(N)$, with no rate. The problem's first demand, to estimate $f(N)$, is not addressed: the manuscript gives no upper bound of a specific order, and the author wrote in the claim's thread on 14 July 2026 that the draft does not determine the order of magnitude of $f(N)$. The best bounds in hand stay those on the problem page: $N/(\log N)^{1-1/e+o(1)}\ll f(N)\le(0.7845+o(1))N$ as forum sketches, and, as the site's commentary records them, $(\log\log N)^{\omega(N)}N/\log N\ll f(N)\le(221/225+o(1))N$ for some $\omega(N)\to\infty$, the lower bound Weisenberg's sharpening of the published Abbott--Gardner bound $(1-o(1))N\log\log N/\log N$.

Argument, as the claimant describes it. Three distinct integers with equal pairwise least common multiples are $txy,txz,tyz$ with $x,y,z$ pairwise coprime (Lemma 2.1), so for squarefree integers a forbidden triple is three prime supports with equal pairwise unions. Positive density is first reduced to a finite set of primes (Proposition 3.1, the same reduction the site's thread uses for its computer-assisted upper bounds), then to a squarefree capacity; the supports are then arranged in an $\mathbb F_3$-cube whose coordinates are disjoint blocks of primes, so that every affine line of the cube is a forbidden triple and the Ellenberg–Gijswijt cap-set bound gives an exponential saving. The cutoff $n\le N$ forces the cube's vertices to have nearly equal products; the manuscript meets this with separated prime bands, a first-moment count of balanced choices, a second-moment bound on the bias that conditioning on balance introduces, and an average over spare bands, and it completes the proof in its Section 8. A companion exact-arithmetic verifier covers its finite checks. Sections 4 to 8, the core of the argument, are outside this page's basis.

Claimant. Shouqiao Wang, whose title page gives the affiliations Columbia University and Multiscalar Intelligence; the tab's submission declares that the claim was produced using an AI system (GPT-5.6 Sol), and the manuscript itself carries no statement about AI use. The author posted a Lean directory in the same repository on 27 July 2026 (linked above at the pinned commit); the corpus has not built or audited it, so it is not counted as formalized. The manuscript's repository carries an MIT license for the repository as a whole.

Standing. Claimed, partial. The claim was submitted to the site's proof-claims tab on 14 July 2026 as a full proof. In the tab's thread the same day, a reader observed that the manuscript answers only the second part of the question and leaves the main question, estimating $f(N)$, open; the author agreed to having treated $f(N)=o(N)$ as the whole problem and offered to withdraw the submission; the site's curator kept it and relabeled it a partial proof, remarking that whether the main question is to estimate $f(N)$ or to prove $f(N)=o(N)$ is a matter of taste. The site's label is OPEN, its commentary (last edited 29 April 2026) does not mention the manuscript, and no arXiv version, refereed publication, independent review or written dispute of the argument was found (tab and thread as of 2026-10-06; the problem page's search scope of 2026-09-18 records the other routes). The corpus has not reviewed the claim.

Depends on. No page of this wiki.