Wang 2026 proposed complete solution erdos problem 536

../

theorem_1_1: The manuscript's claim that every set of positive upper density contains an lcm triangle, with its pair-product lemma and its own proof outline; an unreviewed claim, not an accepted result.

---

Shouqiao Wang, A Proposed Complete Solution to Erdős Problem 536. Preprint (github.com/ShouqiaoW/erdos) (2026).

The retained folder-name PDF is a 33-page pdfTeX manuscript with a clean text layer, undated in its text (file metadata 22 July 2026), whose title page carries the author's two affiliations, Columbia University and Multiscalar Intelligence. Provenance: retained from the repository's survey download set of September 2026; its bytes are identical (equal git blob identity) to 536/paper.pdf in the repository github.com/ShouqiaoW/erdos, last changed in that repository's commit of 22 July 2026 and unchanged at its head commit d28713ac of 2 August 2026 (listing read through the GitHub API on 2026-09-18; the directory also holds the manuscript's source, a Lean directory and a Python verifier, none read); 509,226 bytes (the Git LFS pointer records the digest). The finite-prime envelope is Proposition 3.1 in this version; a site comment of 7 September 2026 cites it as Proposition 2.5, so another version exists. Standing: an unreviewed claim. It was submitted to the site's proof-claim tab for Problem 536 on 14 July 2026, where the tab labels it a partial proof claim; the site's label stays OPEN and its commentary does not mention it; no arXiv version, refereed publication or independent review was found on 2026-09-18. The tab's submission declares that the claim was produced using an AI model (GPT-5.6 Sol); the manuscript itself carries no statement about AI use. The file prints no notice of its own on pp. 1--2 or 32--33; the repository holding it carries a LICENSE file that GitHub shows as "MIT license", the MIT License for the repository as a whole (https://github.com/ShouqiaoW/erdos, read 2026-10-02), and whether the author meant it to cover the manuscript text is not stated.

Read status: claims checked for Theorem 1.1 (p. 1), Lemma 2.1 (p. 2, whose proof was read in full), Propositions 2.2, 2.3 and 3.1 (pp. 3--4) and the closing of the proof (p. 32) in the text layer and on the page images; Sections 4--8, the core of the argument, were not read. Result page: theorem_1_1 (a claim page).

Let f(N) be the largest size of a subset of {1,...,N} containing no three distinct a, b, c with lcm(a,b) = lcm(a,c) = lcm(b,c). Theorem 1.1 claims f(N) = o(N), so every set of positive upper density contains such a triple, answering Erdős's question. The proof rests on Lemma 2.1, the pair-product form: three distinct integers form such a triple exactly when, in some order, they are txy, txz, tyz with t, x, y, z positive and x, y, z pairwise coprime; this is applied at four scales along the chain positive density implies a finite-prime envelope implies a squarefree moving-prefix capacity implies balanced pair-product cubes implies a cap-set saving. The technical core is a five-state prime-band coupling giving a first-moment bound P(B_T) >> w^2 and a matching second-moment bound E[P(B_T | S)^2] << w^4, whose L2 control is converted by averaging over alternative bands into the L1 marginal flatness the moving-prefix argument needs. External inputs are only standard prime-number estimates, Brun-Titchmarsh, and the polynomial-method cap-set bound, with finite checks in a companion Python verifier. The manuscript does not mention Problems 535 or 857 (its text was searched for their numbers and for gcd and sunflower wording on 2026-09-18).

Source: https://github.com/ShouqiaoW/erdos/tree/main/536.

Bears on. #536: Theorem 1.1 claims the answer $f(N)=o(N)$ to the site's question; recorded there as an unreviewed claim.

Results to transcribe.

• Theorem 1.1 (claim page): Claimed f(N) = o(N) for the largest subset of [N] with no three distinct elements having equal pairwise least common multiples.
• Lemma 2.1: Pair-product form: three distinct integers have equal pairwise lcms iff, in some order, they are txy, txz, tyz with x, y, z pairwise coprime.
• Band moment estimates (Sections 5-6): Five-state prime-band coupling giving P(B_T) >> w^2 and E[P(B_T|S)^2] << w^4, yielding uniform L2 control of the conditioned root density.