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The claim. For $1\le a_1<\dots<a_k\le x$ and $1\le b_1<\dots<b_l\le x$ with all products $a_ib_j$ distinct, $kl<c\,x^2/\log x$ for an absolute constant $c$; this is the statement of Problem 490 with $x$ for $N$ and settles it in the affirmative. P. Erdős and A. Szemerédi, On multiplicative representations of integers, J. Austral. Math. Soc. Ser. A 21 (1976), no. 4, 418--427, received 1 December 1974 (the date this page is named by, the earliest dated record of the claim), published June 1976; Theorem 1, printed p. 421, paged as Theorem 1 of Erdős and Szemerédi (1976). The paper presents it as a simpler proof of Szemerédi's theorem, paged as Szemerédi's claim, reusing many of its ideas: primes associated with $A$ or $B$ by dividing a positive proportion of the members, a passage to subsequences of at least half the size, distinct pairs of quotients over a dyadic block of primes associated with both, and an upper count by Brun's sieve and Mertens's theorem. It was followed here for its structure, not checked step by step. The same paper conjectures the sharper bound $(1+o(1))x^2/\log x$, which a forum construction of 7 September 2026 reports to be false; that conjecture is not the problem.

Acceptance. Refereed: the journal publication cited above. The site's commentary attributes the problem's proof to Szemerédi's 1976 paper and does not cite this one; a thread comment of 17 May 2026 links it from the Rényi Institute's Erdős archive as the shorter proof.

Depends on. Nothing in this wiki: the theorem is proved within the paper, whose card is linked above.