Problem 662

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claims/: The 2 claim pages of Problem 662, one per claimant's result; the problem's standing derives from them.

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Statement. Consider the triangular lattice with minimal distance between two points $1$. Denote by $f(t)$ the number of distances from any points $\leq t$. For example $f(1)=6$, $f(\sqrt{3})=12$, and $f(3)=18$.

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that $d(x_i,x_j)\geq 1$ for all $i\neq j$. Is it true that, provided $n$ is sufficiently large depending on $t$, the number of distances $d(x_i,x_j)\leq t$ is less than or equal to $f(t)$ with equality perhaps only for the triangular lattice?

In particular, is it true that the number of distances $\leq \sqrt{3}-\epsilon$ is less than $1$?

Formulation. The site's commentary says that this wording, essentially verbatim from [Er97e], does not make sense as written; it fixes no intended form, and no source supplies one, so the wording stands and the standing judges it under each reading. Here $f(t)$ is the number of points of the triangular lattice within distance $t$ of a lattice point: $f(t)=6$ for $1\le t<\sqrt3$, $f(\sqrt3)=12$, $f(2)=18$ and $f(3)=36$, so the printed example $f(3)=18$ is wrong, and $18$ is $f(2)$. "The number of distances $d(x_i,x_j)\leq t$" has two readings, and both fail. Counted over pairs, a rhombic patch of $m^2$ triangular-lattice points has $3m^2-4m+1$ pairs at distance $1$, so for every $t\geq1$ the number of pairs at distance at most $t$ exceeds $f(t)$ once $m$ is large. Counted per point, a regular heptagon of side $1$ with its center, padded with far-away points to make $n$ as large as required, has a point with seven neighbors within distance $t$ for $1/(2\sin(\pi/7))<t<\sqrt3$, where $f(t)=6$. The weakest comparison, the average number of neighbors $2E_t(X)/n$ of an $n$-point set $X$ against $f(t)$, also fails: $m\times m$ patches of the square lattice have average degree tending to $8$ at every $t$ in $[\sqrt2,\sqrt3)$, where $f(t)=6$. A failure of the average is a failure of both readings, since some point then has more than $f(t)$ neighbors. Chojecki's note records the three failures (its display (1), Proposition 1 and Theorem 2(c)). The final clause, with either count, fails for $0<\epsilon\leq\sqrt3-1$ in any set with a pair at distance $1$. The smallest-distinct-distances question under Known Results is not a reading: it replaces the threshold $t$ by the rank of a distance occurring in the set and divides the count by $n$, neither of which the wording does. The stronger conjecture the site quotes from [Er97e], which compares the distances below the $n$th lattice distance $t_n$ with $f(t_n)$, and the repaired questions under Known Results are variants with their own answers.

The site labels the problem OPEN, and its commentary says the wording, essentially verbatim from [Er97e] p. 532, "does not make sense as written; there must be at least one typo", invites suggestions about what it intends, and calls the stronger shell conjecture it quotes "nonsense interpreted literally". The curator has proposed no form, has not replied in the discussion thread and has not commented on the proof claim, so no ruling of the curator's fixes a Statement for this page to follow, and the label describes a question the site does not state. Both claimants report that the 1997 print matches the site's wording: Chojecki's note calls the site's text essentially verbatim and places the corrupt data (the value $f(3)=18$, which is $f(2)$, and the shell list $1,\sqrt3,3,5$, which omits $2$ and $\sqrt7$) in Erdős's print, and Snyder's write-up reconstructs 943 consecutive characters of p. 532 from National Diet Library full-text snippets (NDL PID 10996926) and reports the same elements in the same order: the lattice and $f(t)$, a separated set, the comparison of its short distances with $f(t)$, the equality case, the $\sqrt3-\epsilon$ clause and the shell conjecture. The defect is therefore Erdős's own print, which this corpus has not read; the claimants' reconstructions are theirs, not a reading of the source here. With no form fixed by the curator or by a source, the wording stands and is judged under each counting reading, and every reading fails by direct computation (above). The page therefore departs from the site's label: following OPEN would state as open a wording that is refuted under every reading, and the curator has not ruled on any other. The repairs the claimants propose are variants with their own answers and are not counted: Chojecki's normalized threshold question $M(t)=f(t)$, true for $1\le t<\sqrt2$ and false on $[\sqrt2,\sqrt3)$, and Chojecki's two-smallest-distances reading, which the note identifies with Vesztergombi's theorem $m_1+m_2\le6n$; Snyder's strict-shell conjecture, refuted at squared radius $300$, and Snyder's repaired final clause, fewer than $12$ neighbors below $\sqrt3-\epsilon$, which the write-up proves. If the curator adopts a form, or [Er97e] is read and prints one that differs from the site's wording, the page takes the steps again on that form.

Status. The site labels the problem OPEN and credits no result; its commentary says the wording does not make sense as written and fixes no intended form, so the label describes a question the site does not state. The wording fails under each counting reading (Formulation). Its proof-claims tab carries one claim, submitted 2026-07-15, which the Snyder claim page records, and its discussion thread carries Przemek Chojecki's note of 2026-04-23, which the Chojecki claim page records; each refutes the wording, the site adopts neither, and the frontmatter standing derives from those pending full claims. The page therefore departs from the site's OPEN label, a departure the Formulation explains: following the label would state as open a wording refuted under every counting reading.

Source. erdosproblems.com/662, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #662, https://www.erdosproblems.com/662.

References.

• [Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. The problem is on p. 532. Unread.

Formalization. None recorded.

Current assessment

No source identifies the intended question, and no proposed repair has been compared with the complete printed text of [Er97e]. So no source fixes a form other than the site's wording, which the page judges as printed.

The wording fails under each counting reading, by the direct computations in the Formulation. The standing is derived from the claim pages in claims/. [[problems/distance_problems/E0662/claims/2026_04_23_chojecki|Chojecki's note]] refutes the wording under each reading (its display (1), Proposition 1 and Theorem 2(c)), and [[problems/distance_problems/E0662/claims/2026_07_15_snyder|Snyder's oblique-lattice counterexample]] refutes the average comparison, and with it each reading, at $t=6$. Both are pending full claims, so the problem reads claimed, disproved. The repaired questions each proposes, Chojecki's threshold equality for $1\le t<\sqrt2$, Snyder's strict-threshold counterexample and Snyder's positive answer to a repaired final clause, are variants and are not counted.

OpenAI's release of 23 September 2026 claims universal optimality of the triangular lattice, with a Lean formalization: at fixed density it minimizes the energy of every completely monotone function of the squared distance, as its preprint Universal optimality of the triangular lattice states; the library's intake card records the manuscript. That theorem does not bear on this problem: a threshold count of distances is a step function of the distance, not a completely monotone potential, and the problem fixes a minimum separation rather than a density, so the oblique-lattice counterexample below is consistent with it. The release is recorded here for that reason and not as progress.

Known Results

Two mathematically different variants occur in the literature and the discussion; the discussion also suggests a densest-packing variant, which Fejes Tóth's theorem on the hexagonal packing answers yes. The two variants are stated separately below.

1. Threshold counts. For a finite planar set $X$ with minimum separation at least $1$, let $E_t(X)$ count unordered pairs at distance at most $t$. A possible comparison with the triangular-lattice neighbor count uses the average degree $2E_t(X)/|X|$. The April 2026 note Reconstructing a corrupted Erdős problem on small distances defines

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

It claims $M(t)=6$ for $1\leq t<\sqrt2$, and $M(t)\geq8$ for $\sqrt2\leq t<\sqrt3$. The equality is a claim about a repaired threshold question, a variant; the lower bound $M(t)\geq8$ also refutes the Statement (Formulation). The claim page records both. The note's assertion that its interpretation is historically definitive is not adopted here.

2. Smallest distinct distances. Let $d_1<d_2<\cdots$ be the distinct distances of an $n$-point set, and let $m_i$ count unordered pairs at distance $d_i$, assuming at least two distance values occur. Vesztergombi's Theorem on p. 100 proves $m_1+m_2\leq6n$. This counts the first two occurring distance values, regardless of their numerical size. It does not count every pair below a fixed threshold. The same paper's Theorem on p. 99 gives $m_2\leq5n$, and its hexagonal construction gives $m_2=(24/7)n+o(n)$ along finite patches. Brass (The maximum number of second smallest distances in finite planar sets, Discrete Comput. Geom. 7 (1992), 371--379) proved an upper bound for $m_2$ whose constant $24/7$ is best possible, which the threshold note states as $m_2<24n/7$; so the hexagonal construction is asymptotically extremal. Complete local proof compilations are supplied on those result pages, with explicitly labeled source-local repairs. Those bounded proof scopes and this multiplicity application have been independently checked against the published source; the result pages record Verified at the stated scope, and the [[../library/distance_problems/vesztergombi_1987_bounds_number_small_distances_finite_planar_set/evidence/verify/final_review|final review]] filed with the source retains the report. They do not bear on the Statement's fixed-threshold count or certify any fixed-threshold variant. Csizmadia's publisher abstract reports refinements depending on the ratio of the two smallest distances. Only its abstract is cited; its theorems and proofs are not compiled.

The threshold note's proof is not independently certified here. In particular, its Lemma 3 proof (p. 3 of the linked PDF) introduces a restriction on the projections of two vertices of a convex quadrilateral that is not justified by its stated hypotheses; that argument requires review before adoption. No replacement proof is supplied here. The gap touches only the upper bound $M(t)\le6$; display (1), Proposition 1 and the square-lattice bound of Theorem 2(c), which refute the Statement, are direct computations.

The public discussion records Przemek Chojecki's announcement of 23 April 2026 of this note, which Chojecki writes was obtained with GPT-5.4 Pro. Nat Sothanaphan's reply reports that an automated check raised a mathematical issue and objections to its historical presentation. These comments do not establish acceptance of either repair.

On 15 July 2026, Colin Snyder submitted a claim with oblique-lattice patches: a one-separated lattice with more points than the triangular lattice within radius $6$, and a $365\times365$ patch $X$ of it with $2E_6(X)>126|X|$ for the normalized closed-threshold comparison, with a strict-threshold counterexample and a repaired final clause beside it. The claim page records the claim, its Lean archive and the absence of acceptance evidence. Its closed-threshold counterexample refutes the average comparison, and so each counting reading of the Statement, at $t=6$; its strict-threshold counterexample and its repaired final clause are variants.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• openai_2026_atomic_certificate_triangular_lattice_universal_optimality
• openai_2026_atomic_certificate_triangular_lattice_universal_optimality / theorem_1_1
• openai_2026_atomic_certificate_triangular_lattice_universal_optimality / theorem_1_2
• openai_2026_sharp_fourier_certificate_planar_circle_packing
• openai_2026_sharp_fourier_certificate_planar_circle_packing / corollary_5_2
• openai_2026_sharp_fourier_certificate_planar_circle_packing / theorem_1_1
• openai_2026_triangular_minimality_planar_coulomb_renormalized_energy
• openai_2026_triangular_minimality_planar_coulomb_renormalized_energy / theorem_1_1
• openai_2026_universal_optimality_triangular_lattice
• openai_2026_universal_optimality_triangular_lattice / theorem_1_1
• vesztergombi_1987_bounds_number_small_distances_finite_planar_set
• vesztergombi_1987_bounds_number_small_distances_finite_planar_set / construction_pp99_100
• vesztergombi_1987_bounds_number_small_distances_finite_planar_set / theorem_p100
• vesztergombi_1987_bounds_number_small_distances_finite_planar_set / theorem_p99