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Claim. With $U(N)=\max_{A,B\subseteq\{1,\ldots,N\}}F(A,B)$,

$$U(N)\asymp\frac{N^2}{(\log N)^{\delta}(\log\log N)^{3/2}},\qquad \delta=1-\frac{1+\log\log2}{\log2},$$

Theorem 1 of the manuscript "Uniquely Represented Products from Two Subsets of an Initial Interval" (draft dated 26 April 2026, no author named). The upper bound is Ford's multiplication-table theorem (claim page). The new part is the lower bound. For a random half of the large primes $p$ between about $N^{2/3}$ and a constant times $N$, each kept independently with probability $1/2$, $A$ receives the numbers $pr$ with $r$ in $(N/(2p),N/p]$, so $r<p$ and the label $p$ is recovered from $pr$, and $B$ is sifted of multiples of the chosen primes, so products from distinct labels never coincide; within one label, uniqueness of $m=ab$ reduces to $m/p$ having exactly one divisor in a short interval, and Ford's estimates for $H_1(x,y,2y)$, the count of integers up to $x$ with exactly one divisor in $(y,2y]$ (Corollary 2 and Theorem 4 of Ford's paper, on the library card ford_2008_distribution_integers_divisor_given_interval), supply enough such integers, summed over the labels by a prime-sum estimate, to reach the order of the multiplication table. The manuscript's Remark 2 says the theorem fixes the order of magnitude and the logarithmic exponent, not a leading constant. This account rests on the manuscript's abstract, introduction, Section 2 and the construction of Section 4.

Depends on. Ford's bound supplies the upper half of the two-sided estimate. The divisor-interval input of the construction, Corollary 2 and Theorem 4 of Ford's paper, is recorded on no claim page.

Claimant. The manuscript names no author. The forum post of 26 April 2026 by the user Przemek (Chojecki) announces the result as found by GPT-5.5 Pro and links the manuscript; the site's commentary credits the lower bound to GPT-5.5 Pro prompted by Chojecki. The page is filed under the prompter's surname.

Acceptance. Reviewed: the site's curator, Thomas Bloom, who took no part in the claim, accepted the result: the problem page is labeled SOLVED and its commentary (last edited 2 May 2026) states the two-sided estimate, resting the upper bound on Ford and the lower bound on this manuscript; in the thread, Nat Sothanaphan reported on 26 April 2026 that a standard check of the write-up found no issues, and the curator posted on 2 May 2026 a streamlined account of the lower-bound construction, noting that it generalizes Szemerédi's construction of sets with uniquely represented products. There is no refereed version, and the site records no formalized statement. Nothing is independently reviewed by this project.

Scope. Full. The problem asks to estimate the maximum, and the order of magnitude is the estimate the site records as the answer. The claim value is answered: the question asks for an estimate, which is neither a proof nor a disproof of a stated assertion.