---

Claim. The first question of Problem 654, whether $f(n)>(1-o(1))n$, has the answer no. Tony Feng and twenty-three coauthors, Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems, arXiv:2601.22401, first posted 29 January 2026 (v3 on 5 February 2026), present in Section 3.1 a solution the authors attribute to Aletheia, a math research agent built upon Gemini Deep Think, and classify it as a partial AI solution. The paper states the problem in the site's earlier wording, which asked only whether some point must have $(1-o(1))n$ distinct distances; its Remark 3.1 says that the construction fully answers that wording but counts as partial because Erdős and Pach posed a weaker question for points with no three on a line. For an integer $m\ge10$ let $n=4m$ and $K=\{10,\ldots,m+9\}$, and let $S$ consist of the $2m$ points $(0,\pm3^k)$ and the $2m$ points $(\pm2^j,0)$ with $k,j\in K$. A circle meets each axis in at most two points, and a circle through two points of each axis would give $3^{k_1+k_2}=2^{j_1+j_2}$ by the power of the origin, so no four points of $S$ lie on a circle (Lemma 3). From a point of one axis the distances along that axis are integers and those to the other axis are irrational (Lemma 4), so the point sees at most $2m-1$ distances along its own axis and exactly $m$ to the other, at most $3m-1<3n/4$ in all (Theorem 3). Hence $f(n)<3n/4$ for every $n=4m$ with $m\ge10$, and $f(n)>(1-o(1))n$ fails. The source card is feng_2026_semi_autonomous_mathematics_discovery_gemini_case.

Covers. The first question, whether $f(n)>(1-o(1))n$, answered no. The second question, whether $f(n)>(1/3+c)n$ for some $c>0$, and the estimate of $f(n)$ that the site's current statement asks for are not settled. All points of $S$ lie on two lines, so the construction says nothing about points with no three on a line.

Depends on. No page of this wiki.

Standing. The preprint is unrefereed, and the only review recorded is the authors' own expert evaluation. The site's commentary credits Aletheia with disproving the strongest form of the conjecture, but the site labels the problem OPEN, so that commentary is not acceptance. The claim is therefore claimed.