Erdős Problem #503

declared status 'open'. Formalized: yes. What is the size of the largest $A\subseteq \mathbb{R}^d$ such that every three points from $A$ determine an isosceles triangle? That is, for any three points $x,y,z$ from $A$, at least two of the distances $\lvert x-y\rvert,\lvert y-z\rvert,\lvert x-z\rvert$ are equal. Current best: The best upper bound known in general is due to Blokhuis [Bl84] who showed that$$\lvert A\rvert \leq \binom{d+2}{2}.$$Alweiss has observed a lower bound of $\binom{d+1}{2}$ follows from considering the subset of $\mathbb{R}^{d+1}$ formed of all vectors $e_i+e_j$ where $e_i,e_j$ are distinct coordinate vectors. Weisenberg observed in the comments that an additional point can be added to Alweiss' construction, giving a lower bound of $\binom{d+1}{2}+1$. Prize: no. OEIS: A175769. Tags: distances, geometry.

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