Erdős Problem #657
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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Is it true that if \(A\subset \mathbb{R}^2\) is a set of \(n\) points such that every subset of \(3\) points determines \(3\) distinct distances (that is, \(A\) has no isosceles triangles, degenerate ones included) then \(A\) determines at least \(f(n)n\) distinct distances, for some function \(f\) with \(f(n)\to \infty\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/657.Attempts
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Exact elementary/algebraic certificates prove \(\phi(n,3,3)=3,3,5,6\) for \(n=3,4,5,6\); the planar asymptotic question remains open.
PARTIAL