Erdős Problem #959
OPENdatabase updated 2025-08-31
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).
The unordered index pairs, represented uniquely by the orientation `i < j`.
The finite set of distinct distances determined by a configuration.
\(f(d)\): the number of unordered pairs in \(P\) determining distance \(d\).
Let \(A\subseteq \mathbb{R}^2\) be a set of size \(n\) and let \(\{d_1,\ldots,d_k\}\) be the set of distinct distances determined by \(A\). Let \(f(d)\) be the number of times the distance \(d\) is determined, ordered so that \(f(d_1)\geq f(d_2)\geq \cdots \geq f(d_k)\). Estimate \[\max (f(d_1)-f(d_2)),\] where the maximum is taken over all \(A\) of size \(n\) (this is `extremalGap n`).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/959.Attempts
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The authoritative live page is OPEN with no current worker, but it lists 2 claimed partial proofs (submitted 2026-07-15 and 2026-07-21), so the mandatory no-duplication gate applies.
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