Erdős Problem #653
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(x_1,\ldots,x_n\in \mathbb{R}^2\) and let \(R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\}\), where the points are ordered such that \[R(x_1)\leq \cdots \leq R(x_n).\] Let \(g(n)\) be the maximum number of distinct values the \(R(x_i)\) can take. Is it true that \(g(n) \geq (1-o(1))n\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/653.Attempts
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Proved \(g(n)\le n-2\) for every \(n\ge7\) and, by exact nested \(\mathbb Q(\sqrt3)\) constructions, \(g(7)=5\), \(g(8)=6\), and \(g(9)=7\); the asymptotic question remains open.
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