---

Source. T. Feng, T. Trinh, G. Bingham et al., Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems, arXiv:2601.22401v3 (5 February 2026); Section 2.1, the problem as posed and Remark 2.1 on p. 9, the solution on pp. 10--11. The result is unnumbered; the paper's Theorem 1 (p. 10) is the Pach--Sharir incidence bound it quotes. The artifact is identified on the source card.

Read depth. Claims checked: the problem, the assertion and the proof (pp. 9--11) were read in full on the print; the cited incidence theorem was not checked against its source. Nothing here is independently reviewed. A preprint.

Statement

For points $x_1,\ldots,x_n\in\mathbb R^2$ let $R(x_i)=\#\{|x_j-x_i|:j\ne i\}$, with the points ordered so that $R(x_1)\le\cdots\le R(x_n)$, and let $\alpha_k$ be minimal such that for all large enough $n$ some $n$-point set has $R(x_k)<\alpha_k n^{1/2}$ (p. 9). The paper proves (p. 10) that $\alpha_k=\Omega(k^{1/4})$ as $k\to\infty$, and hence that $\alpha_k\to\infty$.

Proof pointer

Fix $k$ and, for large $n$, an $n$-point set with $R(x_k)<\alpha_k n^{1/2}$. The circles centred at $x_1,\ldots,x_k$ through the other points number fewer than $k\alpha_k n^{1/2}$, and each of the $n-k$ remaining points lies on $k$ of them. The Pach--Sharir bound (the paper's Theorem 1, quoted from Pach and Sharir 1998, Theorem 1.1), applied to circles with $k=3$, $s=2$ and exponents $(3/5,4/5)$, bounds the incidences above; dividing by $n$ and letting $n\to\infty$ gives $k\le C(\alpha_k^{4/5}k^{4/5}+1)$ (pp. 10--11). Remark 2.1 (p. 9) states that the agent's original output used wrong exponents $(2/3,2/3)$ from a reference the authors could not find, and an $\alpha_k+\epsilon$ limit step; the displayed solution corrects both.

Dependencies

Pach and Sharir, the incidence bound for curves with $k$ degrees of freedom and multiplicity type $s$ (cited, not held).

Bears on

• Problem 652: the statement answers the question as posed, with the rate $k^{1/4}$; the paper notes (p. 9) that the solution is an immediate reduction to the literature and that a later solution by others uses a theorem of Mathialagan instead.