Problem 327

Suppose $A\subseteq \{1,\ldots,N\}$ is such that if $a,b\in A$ and $a\neq b$ then $a+b\nmid ab$ . Can $A$ be 'substantially more' than the odd numbers?

What if $a,b\in A$ with $a\neq b$ implies $a+b\nmid 2ab$ ? Must $\lvert A\rvert=o(N)$ ?

Source: erdosproblems.com/327

Number theory
Unit fractions

Wiki page

Status

ClaimedAnswered

A full solution has been claimed but not yet accepted. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Open on the site: the label is OPEN (the page shows no last-edited date) and the site marks the problem as not resolvable by a finite computation. The standing derived from the claim pages is claimed, claim answered: one full claim, Della Pietra's AI-assisted manuscript of 29 July 2026 (its page), asserting a positive answer to the first question and an independent negative answer to the second, is pending and accepted by nobody. Second question: Sawin's Theorem 1 (arXiv:2607.15419v1, 16 July 2026) constructs sets of positive density with $a+b\nmid2ab$ for all distinct pairs, a negative answer, filed as a partial claim (its page); the source is an arXiv preprint with no journal acceptance, independent review or citing work found on 2026-09-17, so the claim is pending and derives nothing. No refereed source answers either question.

ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), p. 37 (the site gives no page; Sawin and Della Pietra cite p. 37). Library home: Erdős and Graham 1980.
Sa26: Sawin, W., Sets of unit fractions without two members whose average is a unit fraction. arXiv:2607.15419v1 (16 July 2026, the only version, 9 pages); preprint, no journal record. Theorem 1, p. 1; the AI-use declaration, p. 2. Library home: Sawin 2026.
vD25: van Doorn, W., Two-colouring and density lead to many solutions of $1/x+1/y=1/z$ . Undated GitHub note (2025); Theorem 2, p. 3. Library home: Doorn 2025.
Li26: Liu, Yu Leon, Upper bounds for Erdős Problem #327 and its $k$ -variant. Public four-page manuscript (PDF dated 19 May 2026) linked from the site's thread (comment of 13 May 2026; the certificate script erdos327_cert.py was committed 20 May 2026 per the GitHub API); unrefereed; not filed in the library. Theorem 6, p. 2; Table 1 and the AI-use acknowledgment, p. 4.
DP26a: Della Pietra, D., A positive-density improvement over the odd numbers in Erdős Problem 327. Manuscript in the GitHub repository donalddellapietra/erdos-327-proof: version 1 of 29 July 2026 (14 pages, the release PDF) and the revised version of 30 July 2026 (15 pages, at the repository head), the version cited here; the companion [DP26b], Admissibility variants of Erdős Problem 327: the multipliers $k\ge2$ (16 pages), was added to the repository on 30 July 2026 (dates per the GitHub API, UTC); unrefereed; not filed in the library.
Ko26: Korsky, S., Large sets of integers with no harmonic triples. arXiv:2607.05823v1 (7 July 2026); [Sa26] says its construction answers Korsky's Question 1.2. Not held.
OEIS: Xie, Y., Sequence A384927, The On-Line Encyclopedia of Integer Sequences (2025; entry last modified 21 January 2026, server time): $f_1(n)$ for $n\le5000$ (b-file by S. Kesarwani), with a comment conjecturing that $a(n)/n\to1/2$ ; accessed.

Problem 327

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