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Claim. Mehtaab Sawhney's note On $A\subseteq[N]$ such that $ab+1$ is never squarefree for $a,b\in A$, posted on the author's university page, proves as its Proposition 1.1 that there is an integer $N_0$ such that for every $N\ge N_0$, a set $A\subseteq\{1,\ldots,N\}$ in which $ab+1$ is never squarefree for $a,b\in A$ satisfies

$$\lvert A\rvert\le\#\{n\le N:n\equiv7\pmod{25}\}.$$

Its remark adds a stability statement that the proof gives: for some absolute $\eta>0$, if $\lvert A\rvert\ge(1/25-\eta)N$ and $N$ is large then $A$ lies inside the class $7\bmod25$ or inside the class $18\bmod25$, so equality holds exactly for the class $7\bmod25$ and possibly for the class $18\bmod25$. The note credits earlier progress to van Doorn, Weisenberg and Cambie, whose argument (recorded in the site's commentary) bounds $\lvert A\rvert$ by about $0.105N$ from the fact that every $a\in A$ has $a^2+1$ divisible by $p^2$ for some prime $p\equiv1\pmod4$. The proof splits $A$ into its parts in the two classes and the remainder, and applies two sieve lemmas with error $O(N(\log N)^{-1/2})$ for the count of integers in a residue class avoiding prescribed classes modulo $p^2$, with case analysis on whether the parts contain an even integer; the note says an earlier version needed casework modulo $169$ and $289$ as well. The proposition and its remark are recorded at the depth claims checked, with the outline of Section 3; the proof is not checked here. The note gives no explicit $N_0$. Its Section 4 declares that the proof of Proposition 1.1 was obtained with assistance from ChatGPT 5 Pro, which suggested its Lemma 2.2.

Covers. The question for all sufficiently large $N$: the class $7\bmod25$ attains the maximum, so the answer is yes from some unspecified $N_0$ on. What remains is a finite check of the sizes $N<N_0$, which the note does not quantify, so the problem is resolved except for a finite computation, which is what the site's DECIDABLE label records. The pending partial claim Sothanaphan 2026 makes the threshold explicit at $2.64\cdot10^{17}$ along the same proof structure, and the full claims Li 2026 and Pitchford 2026 assert the exact value $\lfloor(N+18)/25\rfloor$ for every $N\ge1$ and would close that check if either stands.

Acceptance. Reviewed: the site's curator, Thomas Bloom, who neither wrote nor submitted the note, credits it in the commentary (page last edited 6 December 2025; accessed 2026-10-06) with settling the problem for every sufficiently large $N$, states the stability form, and labels the problem DECIDABLE, resolved except for a finite check; the community database lists the label decidable, with a last update of 19 October 2025. That documented acceptance by the site is the evidence listed. Not refereed: the note is an unpublished manuscript with no journal record known here. Not formalized here: the formal-conjectures statement file for the problem (at its commit of 18 September 2026; see the problem page) carries, on its variant erdos_848.variants.asymptotic stating the result for all large $N$, a formal-proof attribute naming the file formal/lean/Erdos/848.lean of the repository The-Obstacle-Is-The-Way/erdos-banger, linked above at the pinned commit of 31 January 2026. Its header says the file formalizes Sawhney's asymptotic theorem and not the full question, names as contributors Raymond Jung and the systems Claude Opus 4.5, GPT-5.2 Pro/xHigh, Gemini 3.0 and Aristotle, and claims a build with no sorry, no native_decide and no axioms; the development was announced in the thread on 28 January 2026 (the account erdosbanger) as made with AI assistance under human orchestration, Sawhney replied that the Lean proof appears to follow the human proof, a comment of 29 January 2026 (the account KStar) questioned its readability and its many uses of native_decide, and the author reported their removal on 30 January 2026. Because the file declares itself a formalization of this note's theorem, it is a link on this page; nothing was built or audited here, so it is not formalized evidence. Nothing is independently reviewed by this project. The result also appears in Section IV.1, by Sawhney and Mark Sellke, of arXiv:2511.16072 (20 November 2025). There it is Proposition IV.1.1, with the stability statement as Remark IV.1.1. The paper is a collection of case studies with GPT-5, and its arXiv record gives no journal reference. The section says the problem was solved by Sawhney and GPT-5 together with online comments of van Doorn, Weisenberg and Cambie. A thread comment of 15 January 2026 suggested the paper as a reference, and the formalization's header cites Sawhney and Sellke (2025).

Date. The note carries no date. Its first announcement on record is a thread comment of 19 October 2025 (the account BorisAlexeev, linked above) congratulating Sawhney on resolving the conjecture for all sufficiently large $n$ and quoting the original problem; the community database's last update for the problem carries the same date. The page is named by that date.

Depends on. No page of this wiki.