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Let $A\subseteq\mathbb{R}^2$ be a set of $n$ points with minimum distance equal to 1, chosen to minimise the diameter of $A$. If $n$ is sufficiently large then must there be three points in $A$ which form an equilateral triangle of size 1?
In fact Erdős believes such a set must have very large intersection with the triangular lattice. This is false for $n=4$, for example a square. The behaviour of such sets for small $n$ is explored by Bezdek and Fodor [BeFo99].
Additional thanks to: Boris Alexeev and Dustin Mixon