---

Claim. The note Reconstructing a corrupted Erdős problem on small distances, dated April 2026 and posted to the site's discussion thread on 2026-04-23 by Przemek Chojecki, who writes that the note was obtained with GPT-5.4 Pro, shows that the printed question of Problem 662 fails under each counting reading. Here $f(t)$ counts the points of the triangular lattice within distance $t$ of a lattice point, so $f(t)=6$ for $1\le t<\sqrt3$. Counted over pairs, the rhombus of $m^2$ triangular-lattice points has $3m^2-4m+1$ pairs at distance $1$ (display (1), p. 2), so for every $t\ge1$ the pair count exceeds $f(t)$ once $m$ is large. Counted per point, Proposition 1 (p. 2): for every $t>1/(2\sin(\pi/7))\approx1.152$, a regular heptagon of side $1$ together with its center, padded with far-away points, is an arbitrarily large one-separated set in which one point has seven neighbors within distance $t$, which refutes the per-point reading for $1/(2\sin(\pi/7))<t<\sqrt3$, where $f(t)=6$. On average, for a finite set $X\subset\mathbb R^2$ whose points are pairwise at distance at least $1$, let $E_t(X)$ count the unordered pairs at distance at most $t$, and let

$$M(t)=\limsup_{n\to\infty}\sup_{\substack{|X|=n\\ \min_{x\ne y\in X}|x-y|\geq1}}\frac{2E_t(X)}{n}$$

be the largest asymptotic average number of neighbors within distance $t$. Theorem 2(c) (p. 3): $M(t)\ge8>f(t)=6$ for $\sqrt2\le t<\sqrt3$, from $m\times m$ patches of the square lattice, which are one-separated and have $4m^2-6m+2$ pairs at distance at most $\sqrt2$. A failure of the average is a failure of the other two readings, since some point then has more than $f(t)$ neighbors. So the main question has the answer no, whichever count is meant.

The note proposes the comparison of $M(t)$ with $f(t)$ as the threshold repair of the printed statement and claims, as the rest of its Theorem 2, that $M(t)=0$ for $0<t<1$ and $M(t)=6=f(t)$ for $1\le t<\sqrt2$. Read as the question whether $M(t)=f(t)$, the repair has the answer yes for $1\le t<\sqrt2$ and no on $[\sqrt2,\sqrt3)$, and its Corollary 5 states that the repaired "in particular" clause about $\sqrt3-\epsilon$ is true for $\epsilon>\sqrt3-\sqrt2$ and false for $0<\epsilon\le\sqrt3-\sqrt2$. The upper bound $M(t)\le6$ below $\sqrt2$ rests on the note's Lemma 3, that a convex quadrilateral whose four sides have length at least $1$ has a diagonal of length at least $\sqrt2$: the graph of pairs at distance at most $t<\sqrt2$ then has no crossing edges, so Euler's formula gives $E_t(X)\le3|X|-6$ (Corollary 4). The repair is a variant of the problem and is not counted.

The note's second repair reads the $\sqrt3$ line as the question on the two smallest distinct distances of a finite set, whose answer it identifies with [[../library/distance_problems/vesztergombi_1987_bounds_number_small_distances_finite_planar_set/theorem_p100|Vesztergombi's theorem]] $m_1+m_2\le6n$ and Csizmadia's refinements; the note presents this as a historical identification and claims no new result there. The note's assertion that this reading is the historically definitive one is not adopted on the problem page.

Submission note. Posted to the site's forum by Przemek Chojecki on 23 April 2026:

I got GPT-5.4 Pro to do a search through literature to find the right statement. There are 2 variants possible, both solved. Here's a note on it.

Depends on. No page of this wiki.

Acceptance. None documented. The site labels the problem OPEN and credits no result. The site's curator has not replied in the thread or ruled on a form, so no site reading exists for the refutation to be measured against other than the wording itself. A reply on the thread on the day of the posting reports that an automated check of the note raised one minor mathematical issue and objected to its historical presentation. The problem page records a gap in the proof of Lemma 3: it writes the two vertices off the diagonal $AC$ as $(x,h)$ and $(y,-k)$ with $0\le x,y\le p=|AC|$, a restriction on their projections that the stated hypotheses, a convex quadrilateral with sides at least $1$, do not justify; the lemma, and with it the upper bound of Theorem 2(b), needs a repaired argument before adoption. The refutation of the printed question rests only on display (1), Proposition 1 and the square-lattice bound of Theorem 2(c), which are direct computations unaffected by that gap. Colin Snyder's claim of 2026-07-15 (claim page) addresses the same main question at larger radii, where Snyder finds the triangular lattice is not extremal, and credits this note with the prior work on the problem. The claim is therefore claimed.