Erdős Problem #670
OPENdatabase updated 2026-04-16
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A finite set \(A\) of points has *separated distances* if all its pairwise distances differ by at least \(1\): for any two different unordered pairs \(\{a, b\} \neq \{c, e\}\) of distinct points of \(A\), we have \(\lvert d(a, b) - d(c, e)\rvert \geq 1\).
Let \(A\subseteq \mathbb{R}^d\) be a set of \(n\) points such that all pairwise distances differ by at least \(1\). Is the diameter of \(A\) at least \((1+o(1))n^2\)? The quantifiers in the source are ambiguous. Following the remarks on erdosproblems.com, this is the regime where \(d\) is fixed and the \(o(1)\) term tends to \(0\) as \(n \to \infty\), at a rate which may depend on \(d\). The case \(d = 1\) was proved by Erdős (`Erdos670.erdos_670.variants.dim_one`), so the question is open for \(d \geq 2\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/670.Attempts
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Exact finite progress—proved by a standalone exact certificate that for every d>=3 the minimum five-point diameter is 10+t_*=10.097819085934546..., with an explicit R^3 Gram construction; the fixed-d
PARTIAL