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Claim. There is a set of positive integers of density one whose increasing enumeration has distinct products on distinct consecutive blocks; the answer to the question is yes.

Submission note. Posted to the site's forum by Przemyslaw Chojecki on 17 January 2026:

I think I have the final proof. You can check the preprint here. However, I'm not really happy with it because it's pretty tedious. I plan on trying to formalize it now in Lean with Aristotle and simplify. If you have any ideas on how to make it simpler, I'd be grateful!

Here's why it's tedious: when you try to construct a sequence in question, you note that greedy sequence looks promising but leaves no control, so the next step is to go dyadic and try something like stepwise-greedy dyadic construction. This runs into a wall too (but maybe manageable by some combinatorics - this is my hope for making it simpler). So eventually I settled for a dyadic Moser-Tardos construction (Lovasz local lemma), but that comes with a lot of bookkeeping.

The final construction is:

• reduce preliminary cases (sprinkling)
• use geometry to treat logarithmic regime (Castryck-Cluckers-Dittmann-Nguyen, Bilu-Tichy and Hajdu-Tijdeman results)
• let Moser-Tardos framework deal with the rest of the cases

Any feedback welcome!

The result. P. Chojecki posted a write-up presented as a proof in the site's thread on 17 January 2026 ("I think I have the final proof"), the PDF erdos421-long.pdf served by the author's organization, after a write-up of 16 January (erdos421.pdf) that by its own account pushed the determinant method into the regime of block length of order $\log N$ without settling the problem. The argument follows the route of Tao's thread comment of 18 October 2025: a preliminary sprinkling reduces the cases; collisions of block products are integral points on split-product curves, counted in the logarithmic regime through the dimension-growth bounds of Castryck, Cluckers, Dittmann and Nguyen, with the degenerate curves classified by the theorems of Bilu and Tichy and of Hajdu and Tijdeman; and a dyadic deletion chosen by the algorithmic Lovász local lemma (Moser--Tardos) removes the remaining collisions. The author's thread comments describe the January work as produced in a long conversation with GPT-5.2.

Why it is rejected. On 18 January 2026 a thread commenter listed four problems: the split-product equation did not match the collision it encoded (it took the wrong endpoint of the later block, so its factors carried the wrong sign), the forward argument on the $y$ side rested on the same confusion, and two lemmas of the deletion step counted kept points as if the sprinkled set contained every integer of an interval; a second commenter reported further issues flagged by an AI check (ChatGPT). On 19 January the author acknowledged that some of these were real errors and posted a modular rewrite, erdos421-mod.pdf, with the endpoint corrected, the regularity of the sprinkled set made explicit and the combinatorics rewritten; on 20 January two commenters reported that AI checks (ChatGPT; Gemini 3) still found gaps in the rewrite, in the local-lemma step among others, and no reader confirmed it. Neither version was withdrawn in the thread. On 13 July 2026 the author posted a different, shorter proof on a gap-greedy construction, which the author presented as the final one and which is the result recorded on Chojecki's accepted claim; the January argument is superseded by its author and recorded here as rejected on the acknowledged errors. Nobody is known to have reviewed either January version, and neither is held in this corpus.

Depends on. No page of this wiki.