Erdős Problem #502
SOLVED LEANdatabase updated 2026-02-02
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
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, \[\# \{ \lvert x-y\rvert : x\neq y\in A\} = 2.\] Asked to Erdős by Coxeter. Bannai, Bannai, and Stanton [BBS83] have proved that \[\lvert A\rvert \leq \binom{n+2}{2}.\] A simple proof of this upper bound was given by Petrov and Pohoata [PePo21]. The exact maximum is not known in general: a lower bound of \(\binom{n+1}{2}\) follows from the construction of Alweiss (see [503]).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/502.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanyes
OEISA027627
Original pageerdosproblems.com/502
Attempts
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