What is the size of the largest $A \subseteq \mathbb{R}^n$ 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 $|x - y|$, $|y - z|$, $|x - z|$ are equal.

When $n = 2$, the answer is 6 (due to Kelly [ErKe47] - an alternative proof is given by Kovács [Ko24c]).

[ErKe47] Erdős, Paul and Kelly, L. M., Elementary Problems and Solutions: Solutions: E735. Amer. Math. Monthly (1947), 227-229. [Ko24c] Z. Kovács, A note on Erdős's mysterious remark. arXiv:2412.05190 (2024).