Problem 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,\[\# \{ \lvert x-y\rvert : x\neq y\in A\} = 2.\]
Categories:
Geometry Distances
Progress
Asked to Erdős by Coxeter. Erdős thought he could show that $\lvert A\rvert \leq n^{O(1)}$, but later discovered a mistake in his proof, and his proof only gave $\leq \exp(n^{1-o(1)})$.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].
Shengtong Zhang has observed that a simple lower bound of $\binom{n}{2}$ is given by considering all points with exactly two coordinates equal to $1$ and all others equal to $0$. In fact, since these points are on a hyperplane, a suitable projection gives an improved lower bound of $\binom{n+1}{2}$ (see the construction of Alweiss given in [503]).
Source: erdosproblems.com/502 | Last verified: January 15, 2026