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

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]).