OPEN
This is open, and cannot be resolved with a finite computation.
Let $A$ be a set of $n$ points in $\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$. Is the diameter of $A$ $\gg n$?
Perhaps the diameter is even $\geq n-1$ for sufficiently large $n$. Piepmeyer has an example of $9$ such points with diameter $<5$. Kanold proved the diameter is $\geq n^{3/4}$. The bounds on the distinct distance problem
[89] proved by Guth and Katz
[GuKa15] imply a lower bound of $\gg n/\log n$.
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Additional thanks to: Shengtong Zhang, Boris Alexeev, and Dustin Mixon
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T. F. Bloom, Erdős Problem #100, https://www.erdosproblems.com/100, accessed 2026-01-16
