DISPROVED LEANdatabase updated 2025-12-27
The statement as posed is false. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`A` is a set of equidistant points on a line: there are a point `a` and a non-zero direction `v` such that `A` consists of the points `a + i • v` for `0 ≤ i < #A`.
`A` is a set of equidistant points on a circle: there are a centre `c`, a radius `r > 0`, an initial angle `θ` and a non-zero angular step `α` such that `A` consists of the points of the circle of centre `c` and radius `r` at the angles `θ + i * α` for `0 ≤ i < #A`.
Let \(A\subset \mathbb{R}^2\) be a finite set of size \(n\), and let \(\{d_1,\ldots,d_k\}\) be the set of distances determined by \(A\). Let \(f(d)\) be the multiplicity of \(d\), that is, the number of unordered pairs from \(A\) of distance \(d\) apart. Is it true that \(k=n-1\) and \(\{f(d_i)\}=\{n-1,\ldots,1\}\) if and only if \(A\) is a set of equidistant points on a line or a circle? Erdős conjectured that the answer is no, and other such configurations exist. This was proved by Clemen, Dumitrescu, and Liu [CDL25], who observed that equidistant points on a short circular arc on a circle of radius \(1\), together with the centre, are also an example. The classification is asked for all sufficiently large \(n\). Small exceptions such as \(\{(0,0), (1,0), (0,1), (0,-1)\}\) exist, so the negative answer asserts counterexamples of arbitrarily large size, as in [CDL25].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/958.
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