Erdős Problem #655
ambiguous statement
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).
A collection \(x_1, \dots, x_n\in\mathbb{R}^2\) is _valid_ if no circle whose centre is one of the \(x_i\) contains three other points.
Let \(x_1,\ldots,x_n\in \mathbb{R}^2\) be such that no circle whose centre is one of the \(x_i\) contains three other points. Are there at least \[(1+c)\frac{n}{2}\] distinct distances determined between the \(x_i\), for some constant \(c>0\) and all \(n\) sufficiently large? The answer is **no**: as Zach Hunter observed, the regular `n`-gon (`n` points equally spaced on a circle) is valid and determines only `⌊n/2⌋ < (1+c)n/2` distinct distances, for every `c > 0`. (In the spirit of related conjectures of Erdős and others, presumably some kind of assumption that the points are in general position was intended; see `erdos_655.variants.general_position`.) The disproof — the regular `n`-gon construction together with its supporting lemmas — is formalised at the linked commit.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/655.Attempts
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The literal live-page formulation is false and its exact sharp minimum is \(D(X)=\lfloor n/2\rfloor\), attained for every \(n\) by the regular \(n\)-gon; historically intended no-four-concyclic pinned
PROVED