Problem 536

Let $f(N)$ be the largest size of $A\subseteq \{1,\ldots,N\}$ with the property that there are no distinct $a,b,c\in A$ such that

[a,b]=[b,c]=[a,c],

where $[a,b]$ denotes the least common multiple.

Estimate $f(N)$ - in particular, is it true that $f(N)=o(N)$ ?

Source: erdosproblems.com/536

Number theory

Wiki page

Status

Open

No claim settles this problem.

Open, the site's label (page last edited 29 April 2026; proof-claims tab as of 2026-10-06). No proof or disproof of $f(N)=o(N)$ accepted by the site or by a refereed venue was found in the search whose scope the Current assessment records. In hand: the lower bound $f(N)>(1-\varepsilon)N\log\log N/\log N$ of Abbott and Gardner (Abbott and Gardner (1967)); Erdős's theorem that the four-element analog fails, $F(4,x)>cx$ (1970); a forum lower bound $(\log\log N)^{\omega(N)}N/\log N$ with $\omega(N)\to\infty$ and a forum upper bound $(221/225+o(1))N$ , both accepted into the site's commentary; a chain of later forum upper bounds down to $(0.7845+o(1))N$ (September 2026), among them Saturnino's $(43/48+o(1))N$ (its claim page (Saturnino, 2026)), two with Lean developments their posters describe (Kitamura, Logsdon) and the last with a Lean formalization of its reduction only; and Wang's manuscript of July 2026, hosted on GitHub and submitted to the site's proof-claim tab, which claims $f(N)=o(N)$ and which the site's curator relabeled a partial claim because it does not estimate $f(N)$ . That claim is recorded on its claim page (Wang, 2026), pending and unreviewed; the forum bounds with claim pages are pending too, and Abbott and Gardner's accepted bound is partial; a partial claim of a problem that lists no parts derives no standing, so the frontmatter stays open. This is a bounded negative finding, not a certificate of openness.

Er64: Erdős, P., On a problem in elementary number theory and a combinatorial problem. Math. Comp. 18 (1964), no. 88, 644--646; the closing paragraph, p. 646, the site's cited origin. Library home: Erdős 1964; result page Erdős 1964, Question (p. 646).
Er70: Erdős, P., Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press (1970), 123--133; the definition of $F(k,x)$ , Theorem 1 and the deduction $F(4,x)>cx$ , printed p. 124; Lemma 1 and the start of the proof, pp. 124--125. Library home: Erdos 1970; result page Erdos 1970, Theorem 1.
Er73: Erdős, P., Problems and results on combinatorial number theory. A Survey of Combinatorial Theory (Fort Collins 1971), North-Holland (1973), Chapter 12, 117--138; the first paragraph of printed p. 124. Library home: Erdos 1973 (the card carries its row for this problem).
Er62: Erdős, P., Számelméleti megjegyzések IV. Extremális problémák a számelméletben, I. Mat. Lapok 13 (1962), 228--255. The site's commentary cites it for the four-element result; problem 14 there (pp. 236--238) is the greatest-common-divisor problem and no least-common-multiple triple appears in the paper (see below). Library home: Erdős 1962; result page Erdős 1962, Problem 14 (pp. 236–238).
AbGa67: Abbott, H. L. and Gardner, B., An extremal problem in number theory. Canad. Math. Bull. 10 (1967), no. 2, 173--177 (received 17 November 1966); display (11) and its proof, pp. 176--177. Read at Cambridge Core. Library home: Abbott and Gardner 1967; result page Abbott and Gardner 1967, Display (11).
Guy91: Western Number Theory Problems, 1991-12-19 & 22, edited by Richard K. Guy, "for mailing prior to 1992 (Corvallis) meeting"; problem 91:01, PDF p. 9 of the 18-page image-only scan at https://westcoastnumbertheory.org/wp-content/uploads/2018/02/wcnt-problems-1991.pdf (the site's link, 13,746,604 bytes; accessed). Library home: Guy 1991 (the card carries the row for this problem).
Wa26: Wang, S., A Proposed Complete Solution to Erdős Problem 536. Manuscript, 33 pages, hosted at github.com/ShouqiaoW/erdos (file 536/paper.pdf, last changed in the commit of 22 July 2026). Theorem 1.1 and Lemma 2.1, pp. 1--2; Proposition 3.1, pp. 3--4. Library home: Wang 2026; claim page Wang 2026, Theorem 1.1 (claimed).

Problem 536

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Proof claims

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