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The claim. The squarefree integers $q_1\cdots q_r$ whose prime factors, in increasing order, satisfy $q_{i+1}>2q_i$ have positive density, and for their part $A$ in $(N/2,N)$ no integer $m$ has three representations $m=pa$ with $p$ prime and $a\in A$. Such an $A$ has size $\gg N$ and contains no $a_1,a_2,a_3$ with $a_1p_1=a_2p_2=a_3p_3$ for distinct primes $p_i$, so the question of Problem 537 is answered no for one fixed $\epsilon>0$ and infinitely many $N$. The construction is Imre Ruzsa's, reported by P. Erdős, Problems and results on combinatorial number theory, A Survey of Combinatorial Theory (J. N. Srivastava et al., eds.), North-Holland (1973), Chapter 12, 117--138, printed p. 124 with display (4.4); paged as the construction page of Erdős (1973). Erdős asserts both steps without proof. The at-most-two-solutions step is a two-line deduction: if $p_1a_1=p_2a_2=p_3a_3$ with distinct primes, then $p_2$ and $p_3$ divide $a_1$, so one exceeds twice the other, while $p_3/p_2=a_2/a_3$ lies in $(1,2)$ because $N/2<a_3<a_2<N$; it was checked here. The positive density is Erdős's assertion, proved in the Lean development linked above from a Chebyshev-type bound. No paper by Ruzsa on the construction is cited by the site or by Erdős, so this page is named by the publication year of Erdős's report, the chapter printing no month or day, with the first day of the year standing in for the unknown day.

Acceptance. Reviewed: the site's curator, Thomas Bloom, who had no part in the construction, accepts it as the disproof, with the label DISPROVED (LEAN) and a commentary that writes out the argument. The formalization link is the external Lean file that the formal-conjectures statement names as its proof, at its revision of 30 June 2026; it names Ruzsa as the informal author and the prover Aristotle and Boris Alexeev as the formal authors, proves the negation of the formal statement with the axiom closure its closing comment records as propext, Classical.choice and Quot.sound, and was neither built nor independently audited here, so it is not acceptance evidence. The chapter's DOI is the one the Crossref record gives; the chapter itself prints none.

Depends on. Nothing in this wiki: the construction and its argument are contained in the cited passage, whose card is linked above.