Erdős Problem #668
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).
Is it true that the number of incongruent sets of \(n\) points in \(\mathbb{R}^2\) which maximise the number of unit distances tends to infinity as \(n\to\infty\)? The actual maximal number of unit distances is the subject of [Erdős Problem 90](https://www.erdosproblems.com/90). Two sets of points are congruent if an isometry of the plane maps one onto the other.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/668.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA385657
Original pageerdosproblems.com/668
Attempts
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Elementary proof gives continuum many incongruent maximizers at n=6; modulo AMP25's exact u(n) theorem the same holds at n=8,9,12,21, while no uniform extremality-propagation lemma is known.
PARTIAL