Problem 502

../

claims/: The 4 claim pages of Problem 502, one per claimant's result; the problem's standing derives from them.

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Statement. What is the size of the largest $A\subseteq \mathbb{R}^n$ such that there are only two distinct distances between elements of $A$? That is,

$$\# \{ \lvert x-y\rvert : x\neq y\in A\} = 2.$$

Statement (corrected). Let $M_2(n)$ be the size of the largest $A\subseteq \mathbb{R}^n$ such that there are only two distinct distances between elements of $A$, that is,

$$\# \{ \lvert x-y\rvert : x\neq y\in A\} = 2.$$

Estimate $M_2(n)$: determine its asymptotic behavior as $n\to\infty$.

Notes. The site's wording asks for the exact size of the largest two-distance set in $\mathbb{R}^n$; write $M_2(n)$ for it. That value is known only for $n\le8$, where Lisoněk [Li97] determines $M_2(1),\ldots,M_2(8)=3,5,6,10,16,27,29,45$, and the general bounds $\binom{n+1}{2}\le M_2(n)\le\binom{n+2}{2}$ ($n\ge3$) leave a gap at every $n\ge9$ where the upper bound is not attained, so the exact question is open. The site labels the problem SOLVED (LEAN), a label it defines as resolved other than by a proof or disproof, and its commentary (page last edited 29 January 2026) records the upper bound of Bannai, Bannai and Stanton [BBS83] with Petrov and Pohoata's proof [PePo21] and the lower bounds $\binom n2$ (Zhang) and $\binom{n+1}{2}$ (Alweiss). Its page for Problem 1089 (last edited 1 February 2026) derives from the same two bounds that $\lim_{d\to\infty}g_d(n)/d^{n-1}=1/(n-1)!$, where $g_d(3)=M_2(d)+1$, and says that the behavior of $g_d(3)$ is the focus of this problem. The curator therefore reads the problem as asking for the asymptotic behavior of $M_2(n)$, which the two bounds settle: $M_2(n)=(\tfrac12+o(1))n^2$, indeed $M_2(n)=n^2/2+O(n)$. The reading follows Erdős's own framing. Erdős's source [Er61], printed p. 244, asks "how many points does one have to have in $n$-dimensional space so that one should be sure to have more than two distinct distances between them" and reports that Erdős had claimed $n^{c_5}$ points suffice, that the proof was wrong, and that corrected it gave only $\exp(n^{1-\varepsilon})$: the question Erdős records is the growth order, not an exact value. The corrected Statement asks for that asymptotic behavior; the change names $M_2(n)$ and replaces "What is the size" by "Estimate $M_2(n)$: determine its asymptotic behavior", and nothing else changes. Under the site's wording the problem is open; under the corrected Statement it is solved, by the upper bound ([BBS83], [PePo21]) and the lower construction, the edge midpoints of a regular $n$-simplex, verified on the library's intro_midpoint_construction page and recorded as a claim page on the site's credit. The exact maximum is a variant with its own, open, answer: known for $n\le8$ [Li97], and at $n=23$ Ge, Koolen and Munemasa's 277-point set [GeKoMu26] exceeds $\binom{24}{2}$. Both results are correct, but they answer the site's wording (the exact maximum), not the corrected Statement (the asymptotic behavior of $M_2(n)$), so neither counts toward the problem's standing, and Lisoněk's claim page is rejected. A thread comment of 21 August 2026 (Zakharov) asks whether the problem should be marked open because of the gap. The site's "(LEAN)" suffix rests on the community formalization of the upper bound only.

Formulation. The exact maximum $M_2(n)$, the site's wording's question, is a variant with its own answer, open in general. Lisoněk [Li97] determines it for every $n\leq8$:

$$M_2(1),\ldots,M_2(8)=3,\ 5,\ 6,\ 10,\ 16,\ 27,\ 29,\ 45.$$

Lisoněk's $29$ points in $\mathbb{R}^7$ exceed $\binom82=28$, and the $45=\binom{10}{2}$ points in $\mathbb{R}^8$ attain the upper bound $\binom{n+2}{2}$, so the gap between the bounds is closed at $n=8$; Ge, Koolen and Munemasa's introduction credits Lisoněk with these values, and the OEIS records the sequence as A027627. The result keeps a claim page, rejected because it answers the exact question of the site's wording, not the asymptotic one of the corrected Statement. Ge, Koolen and Munemasa's arXiv:2504.18110v4 [GeKoMu26] constructs a 277-point two-distance set in $\mathbb{R}^{23}$ with distances $2$ and $\sqrt6$, transcribed in full in [[../library/distance_problems/ge_2026_two_distance_set_277_points_23_dimensions/theorem_1|their Theorem 1 page]]; it improves the generic lower bound at $n=23$ from $\binom{24}{2}=276$ to $277$, but does not close the general gap.

Status. The site labels the problem SOLVED (LEAN), its qualifier resting on the community proof of the upper bound; the label describes the corrected Statement. The bounds $\binom{n+1}{2}\leq M_2(n)\leq\binom{n+2}{2}$ for $n\geq3$ give $M_2(n)=n^2/2+O(n)$ (Bannai--Bannai--Stanton 1983 upper bound, with Petrov--Pohoata's 2021 proof; simplex-edge-midpoint lower construction, credited by the site to Alweiss). The problem lists two parts, the upper bound and the lower bound: the two accepted upper-bound claims in claims/ settle the first and the accepted claim for the lower construction settles the second, so the problem's standing is solved. Its claim value is proved where SOLVED (LEAN) reads as answered, because each part of the corrected Statement is a bound and every settling claim proves its bound. The exact maximum, the question of the site's wording, is not known in general and is a variant under Formulation; Lisoněk's determination for $n\leq8$ is a rejected claim.

Source. erdosproblems.com/502, accessed 2026-09-05. Cite as: T. F. Bloom, Erdős Problem #502, https://www.erdosproblems.com/502, accessed 2026-09-05.

References.

• [Er61] Erdős, Paul, Some unsolved problems, Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221--254, p. 244 (the site's original reference).
• [BBS83] Bannai, Eiichi and Bannai, Etsuko and Stanton, Dennis, An upper bound for the cardinality of an $s$-distance subset in real Euclidean space. II. Combinatorica 3 (1983), 147--152. DOI: https://doi.org/10.1007/BF02579288.
• [PePo21] Petrov, Fedor and Pohoata, Cosmin, [[../library/distance_problems/petrov_2021_remark_sets_few_distances/_index|A remark on sets with few distances in $\mathbb{R}^{d}$]], Proc. Amer. Math. Soc. 149 (2021), 569--571. Preprint: https://arxiv.org/abs/1912.08181.
• [GeKoMu26] Ge, Hong-Jun, Koolen, Jack, and Munemasa, Akihiro, A 2-distance set with 277 points in the Euclidean space of dimension 23, Discrete Comput. Geom. 76 (2026), 1679--1686, DOI: https://doi.org/10.1007/s00454-026-00843-9; arXiv:2504.18110v4 (3 August 2026).
• [ChYu25] Chen, Wei-Chun and Yu, Wei-Hsuan, Bounds on two-distance sets in Euclidean space and Unit Sphere, arXiv:2509.00858 (31 August 2025).
• [Li97] Lisoněk, Petr, New maximal two-distance sets, J. Combin. Theory Ser. A 77 (1997), 318--338, DOI: https://doi.org/10.1006/jcta.1997.2749.

Formalization. Formal Conjectures' 502.lean declares the finite-set upper-bound statement $|A|\leq\binom{n+2}{2}$, with a proof body marked sorry; it does not declare the exact-maximum statement. Its annotation points to the pinned Lean file in Alexeev's lean-proofs repository, which contains the Petrov--Pohoata argument and whose closing axiom report lists only the standard Lean axioms; this corpus has not built it.

Current assessment

The site snapshot accessed 2026-09-05 records the label "SOLVED (LEAN)", which describes the corrected Statement: the mathematical status is solved, since the bounds under Progress settle the asymptotic behavior of $M_2(n)$, and the suffix marks the community Lean proof of the upper bound $|A|\leq\binom{n+2}{2}$ only, as recorded under Formalization above. The exact maximum, the variant under Formulation, is not known in general.

The site's discussion and proof-claims pages record the Coxeter origin, the Bannai--Bannai--Stanton upper bound, the Zhang and Alweiss lower constructions, and no submitted proof claim.

The declaration and the pinned proof link therefore provide formal coverage of the upper bound, while neither determines the exact maximum. Neither is Lean this corpus built and audited, so both are links, not acceptance evidence.

The problem's two parts are settled by accepted claims in claims/. The upper bound is recorded as two accepted partial claims, the [[problems/distance_problems/E0502/claims/1983_06_01_bannai_bannai_stanton|Bannai–Bannai–Stanton theorem]] and [[problems/distance_problems/E0502/claims/2019_12_17_petrov_pohoata|Petrov and Pohoata's independent proof]], each settling the part upper_bound; the lower construction with $\binom{n+1}{2}$ points is [[problems/distance_problems/E0502/claims/2025_08_09_alweiss|Alweiss's construction]], accepted on the curator's credit and settling the part lower_bound. The problem's standing derives from them. The exact maximum for $n\leq8$ is [[problems/distance_problems/E0502/claims/1997_02_01_lisonek|Lisoněk's determination]], a rejected claim, since it answers the site's wording and not the corrected Statement. The 277-point set in $\mathbb{R}^{23}$ recorded under Formulation improves the lower bound at one dimension and bears only on the exact-maximum variant.

Progress

Let $M_2(n)$ be the maximum cardinality of a finite subset of $\mathbb{R}^n$ whose distinct pairwise distances are exactly two. Petrov and Pohoata's Theorem 1.1 gives the general upper bound

$$M_2(n)\leq\binom{n+2}{2}. \tag{1}$$

The standard lower construction is the set of midpoints of the edges of a regular $n$-simplex. The canonical statement and distance check are recorded in [[../library/distance_problems/ge_2026_two_distance_set_277_points_23_dimensions/intro_midpoint_construction|the Ge--Koolen--Munemasa introductory construction]]. It gives $\binom{n+1}{2}$ points and, for $n\geq3$, both of its two distances occur. Consequently

$$\binom{n+1}{2}\leq M_2(n)\leq\binom{n+2}{2}\qquad(n\geq3). \tag{2}$$

Any infinite two-distance set would contain finite subsets of arbitrarily large size, so the finite upper bound also rules out an infinite set. The site's commentary attributes the lower construction to Ryan Alweiss and the earlier $\binom n2$ construction to Shengtong Zhang; the construction has [[problems/distance_problems/E0502/claims/2025_08_09_alweiss|its own claim page]].

Known Results

The upper bound in (1) is the Bannai--Bannai--Stanton theorem, with the complete short proof by Petrov and Pohoata in Theorem 1.1; its essential rank/inertia lemma is in Theorem 1.2.

Lisoněk's exact values for $n\leq8$ [Li97] and Ge, Koolen and Munemasa's 277-point set in $\mathbb{R}^{23}$ [GeKoMu26] bear on the exact-maximum variant and are recorded under Formulation. Ge, Koolen and Munemasa's introduction credits the bound $\binom{n+2}{2}$ for two-distance sets to Blokhuis, whose 1984 CWI Tract (card, Theorem 4.1.1) proves $\binom{n+s}{s}$ for $s$-distance sets; it has no claim page because neither the Tract nor Blokhuis's 1984 proceedings note is a journal publication and the site's remarks do not credit Blokhuis.

Chen and Yu's arXiv:2509.00858 is a defect-bearing source lead for a ratio-dependent spectral bound. Their printed Theorem 3.1 omits the required sign condition on its denominator. The valid undivided inequality and the conditional algebraic consequence, together with the source defect, are recorded in [[../library/distance_problems/chen_2025_bounds_two_distance_sets_euclidean_space_unit_sphere/theorem_3_1|the qualified Theorem 3.1 page]]. It is not used to replace the universal bound (1).

The site's discussion says the broader few-distance question is also recorded as Problem #1089.

The exact value of $M_2(n)$, the variant under Formulation, is therefore unresolved in general. The site's discussion thread in the snapshot accessed 2026-09-05 held no proof exposition or proof claim; one comment, of 21 August 2026, asked about the gap, and another pointed to the existing Lean solution.

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.

• blokhuis_1984_few_distance_sets
• blokhuis_1984_few_distance_sets / theorem_4_1_1
• chen_2025_bounds_two_distance_sets_euclidean_space_unit_sphere
• chen_2025_bounds_two_distance_sets_euclidean_space_unit_sphere / theorem_3_1
• ge_2026_two_distance_set_277_points_23_dimensions
• ge_2026_two_distance_set_277_points_23_dimensions / intro_midpoint_construction
• ge_2026_two_distance_set_277_points_23_dimensions / lemma_2
• ge_2026_two_distance_set_277_points_23_dimensions / theorem_1
• petrov_2021_remark_sets_few_distances
• petrov_2021_remark_sets_few_distances / theorem_1_1
• petrov_2021_remark_sets_few_distances / theorem_1_2