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Chen determines the size asked for in the first question asymptotically. The Theorem (p. 71): with $A_x$ a largest set of positive integers whose pairwise least common multiples are at most $x$ and $B_x$ Erdős's construction, the integers up to $(x/2)^{1/2}$ together with the even integers in $[(x/2)^{1/2},(2x)^{1/2}]$, $|A_x\setminus B_x|=o(\sqrt x)$, hence $|A_x|=|B_x|+o(\sqrt x)=\sqrt{9x/8}+o(\sqrt x)$; the Note after the theorem adds $|B_x\setminus A_x|=o(\sqrt x)$. Every set counted by $g(N)$ is a set counted by $|A_N|$ and conversely, so $g(N)=|A_N|\sim(9N/8)^{1/2}$: the largest set has the size of Erdős's construction to within $o(N^{1/2})$, and the extremal sets nearly coincide with it.

Covers. The size part, the first question, in the asymptotic form in which Erdős stated his conjecture ([Er73], quoted on the problem page under Formulation: $\max k=(1+o(1))\frac{3}{2\sqrt2}n^{1/2}$); the value of $g(N)$ is determined to leading order, so the claim value is answered. Dai and Chen's Theorem of 2006 (Acta Arith. 124, pp. 315--316) sharpens it to $-2\le g(N)-(9N/8)^{1/2}\le45(N/\log N)^{1/2}\log\log N$ for large $N$ and conjectures that the remainder tends to infinity; it has no claim page of its own because it refines this answer and settles nothing further. Not covered: the exact value of $g(N)$, unknown in general; the construction part, the second question, which Chen and Dai's theorem settles in the negative.

Acceptance. Refereed: Acta Arithmetica 84 (1998), no. 1, 71--95, DOI 10.4064/aa-84-1-71-95 (Crossref record accessed). The site's commentary (page last edited 27 December 2025) credits Chen with establishing the asymptotic, but its label DISPROVED attaches to the second question, so the curator's credit is not listed as acceptance of this part. Read depth: claims checked for the Theorem and the Note on p. 71; the proof (pp. 72--95) was not read beyond Lemma 1, and nothing is independently reviewed by this project.

Formalization. The community database (teorth/erdosproblems,) names as the problem's formal status, since 16 September 2026, the submission package of Collin Yuanjie Ren in the repository CollinYuanjieRen/awards (submissions/jsp-000360-cyr/, linked above at the commit the database names). Its README presents the package as a Lean formalization of both questions of the problem: its theorem erdos_441_complete states that $g(N)/\sqrt{9N/8}\to1$ and that $|B_N|<g(N)$ for arbitrarily large $N$; the asymptotic follows this paper, and the definitions and the non-optimality theorem are reused from Boris Alexeev's lean-proofs file, the formalization link on Chen and Dai's page. The README says that the package was prepared with OpenAI Codex assistance, claims no new mathematics, and reports the package's own axiom audit (propext, Classical.choice, Quot.sound) with no sorry. Because it declares itself a formalization of this paper's theorem, it is a formalization link on this page and not a claim of its own. Nothing was built or audited here, so it gives no formalized evidence. The statement file of formal-conjectures, described on the problem page, lists Chen's asymptotic as a variant with proof sorry.

Depends on. No page of this wiki. The proof is self-contained in the paper.

Date. The paper carries a year only; the page name uses the first day of 1998 for want of an issue date.