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What is the size of the largest such that there are only two distinct distances between elements of ? That is,
Let be the size of the largest such that there are only two distinct distances between elements of , that is,
Estimate : determine its asymptotic behavior as .
Source: erdosproblems.com/502
An accepted solution exists. The statement is true.
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 for
give (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 is a rejected claim.
The site's wording asks for the exact size of the largest two-distance set in ; write for it. That value is known only for , where Lisoněk [Li97] determines , and the general bounds () leave a gap at every 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 (Zhang) and (Alweiss). Its page for Problem 1089 (last edited 1 February 2026) derives from the same two bounds that , where , and says that the behavior of is the focus of this problem. The curator therefore reads the problem as asking for the asymptotic behavior of , which the two bounds settle: , indeed . The reading follows Erdős's own framing. Erdős's source [Er61], printed p. 244, asks "how many points does one have to have in -dimensional space so that one should be sure to have more than two distinct distances between them" and reports that Erdős had claimed points suffice, that the proof was wrong, and that corrected it gave only : the question Erdős records is the growth order, not an exact value. The corrected Statement asks for that asymptotic behavior; the change names and replaces "What is the size" by "Estimate : 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 -simplex, verified on the library's Ge 2026, Introductory regular-simplex 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 [Li97], and at Ge, Koolen and Munemasa's 277-point set [GeKoMu26] exceeds . Both results are correct, but they answer the site's wording (the exact maximum), not the corrected Statement (the asymptotic behavior of ), 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.