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Claim. Under the distinct-factor reading of Problem 786, in which the two products are over finite subsets of $A$, Declan Gessel's partial proof claim, submitted to the site's proof-claim tab on 6 September 2026 and declared as produced with GPT-6 Astra (Codex), asserts: if $A\subset\mathbb N$ has natural density $\delta$ and two finite subsets of $A$ whose products agree always have the same number of elements, then $\delta\le7/8$. So no such set has density above $7/8$, and the first question has a negative answer for every $\epsilon<1/8$. The argument, in outline: a prime $p$ is called good when $x$ and $px$ both lie in $A$ for infinitely many $x$; each member of $A$ must have some prime divisor that fails to be good, because otherwise new pairs $(x,px)$ would produce two equal products whose factor counts differ; were the density above $7/8$, one could select finitely many primes that are not good whose reciprocals sum to a number in $[1/2,1]$; each dilate $pA$ meets $A$ in a finite set, and a count of the union of the dilates, with the pairwise intersections controlled through the multiples of $pq$, contradicts the assumed density. The claim is stated for natural density; a set with only an upper density is not covered, and the finite question is left open, as the summary says. The manuscript is a gist pinned to its revision, and the Lean file is pinned to its commit.

Submission note. Posted to erdosproblems.com as a proof claim by Declan Gessel (account declangessel) on 6 September 2026, giving "GPT-6 Astra (Codex)" as the AI used:

We answer part (i) negatively for distinct factors. If A is a set of positive integers with natural density δ, and finite subsets of A with equal products always have equal cardinalities, then δ ≤ 7/8. The finite-set question (ii) is not settled. Call a prime p good if infinitely many x have both x and px in A. Every member of A has a prime divisor that is not good; otherwise fresh pairs (x,px) give equal products with different numbers of distinct factors. Assuming density greater than 7/8, select finitely many such bad primes with reciprocal sum between 1/2 and 1. Their dilates pA overlap A only finitely. Counting their union and bounding pairwise intersections by multiples of pq yields a contradiction. The proof is checked in Lean. The bound concerns natural density, not upper natural density. Notes: Historical qualification: Erdős (1980), p. 114, reports a stronger unpublished negative result of Ruzsa in the subset formulation. The current problem page notes possible confusion with the repetition-allowed version. This is an independently derived, machine-checked argument; no priority over Ruzsa is claimed.

Covers. The first question of the problem page's Statement (precise), the distinct-factor reading: no set of natural density above $7/8$ has the property, so the answer is no for $\epsilon<1/8$. Because the distinct-factor property is the weaker condition on $A$, the bound holds under the repetitions-allowed reading as well, for which the site's commentary already attributes a negative answer to Theorem 2 of [ERS73] (see the problem page's Formulation). Not covered: the second, finite question and sets with only an upper density. The later full claim Li 2026 asserts a stronger bound of $1/2$ in logarithmic density and a negative answer to the finite question; its note credits this claim as the earlier answer to the first question.

Acceptance. None on record. The site's label is OPEN (page last edited 11 April 2026, before the claim; proof-claim tab accessed 2026-10-06), the claim has no comments, and no publication or named review was found. The claim's own note says that Erdős's 1980 survey reports a stronger unpublished result of Ruzsa in this setting and that no priority is claimed. The Lean file is not listed as evidence: it has not been built or audited in this corpus. A partial claim derives nothing for the problem's standing.

Depends on. No page of this wiki.