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Claim. Let $g(n)$ be the largest size of a set $A\subseteq\{1,\ldots,n\}$ whose $2^{|A|}$ subset products are all distinct (the paper's $f(N)$). Theorem 1.3 states that

$$g(n)=\pi(n)+\pi(n^{1/2})+O(n^{5/12}).$$

Since $n^{5/12}=o(n^{1/2}/\log n)$, this gives the inequality the problem asks for with room to spare, so the answer is yes; the paper introduces the theorem as the affirmative answer to the site's question and cites the site. Theorem 1.4 gives the lower bound $g(n)\ge\pi(n)+\pi(n^{1/2})+\tfrac13\pi(n^{1/3})-O(1)$, which exceeds the primes-and-prime-squares example and refutes Erdős's 1980 conjecture that $g(n)=\pi(n)+\pi(n^{1/2})+\pi(n^{1/4})+\pi(n^{1/7})+\cdots$; that conjecture is not the problem's question. The source is R. Raghavan, Sharp bounds for sets with distinct subset products, Acta Math. Hungar. 177 (2025), no. 2, 363--377, DOI 10.1007/s10474-025-01578-4 (published online 25 December 2025); arXiv:2501.02695, v1 of 6 January 2025, v2 of 26 February 2026 (13 pp., the version read; no file is held). The displayed form is v2's: v1 proved the upper bound with the error $O(n^{5/12+o(1)})$, which already answers the question, and v2 sharpens it to $O(n^{5/12})$, its acknowledgment crediting Csaba Sándor with the observation; the site's commentary prints the v1 form. Library pages: Theorem 1.3 and Theorem 1.4 on the source card.

Formulation. The site prints the error term with an undefined $x$; the thread's one comment (15 May 2026) calls it a typo for $n^{1/2}/\log n$, and $x=n$ is the only reading, matching Erdős's displays of 1969 and 1970 from which the site's formula comes. The theorem answers the question so read.

Acceptance. Refereed: the paper appeared in Acta Mathematica Hungarica (Crossref). Reviewed: the site's curator, Thomas Bloom, labels the problem PROVED with the note that the answer is yes, the page last edited 6 April 2026, and the curator's commentary credits the paper with both bounds and with the disproof of the 1980 conjecture; the page, accessed 2026-09-18 and 2026-10-07, shows one comment, no proof claim and no exposition. Not formalized: formal-conjectures holds no file for the problem (18 September and 7 October 2026), and the community database records no formal proof. The statements of Theorems 1.3--1.6 are checked in arXiv v2; the proofs (Sections 2--5) were not checked in this repository, the journal text was not compared with the arXiv version read, and this page rests on no review of its own.

Scope. Full for the site's statement. The exact lower-order term of $g(n)$ beyond $\pi(n)+\pi(n^{1/2})$ remains open between $\tfrac13\pi(n^{1/3})-O(1)$ and $O(n^{5/12})$; the problem does not ask for it.

Depends on. No page of this wiki.