For which n can n points in general position have the i-th distance occur exactly i times? (Erdős #217)
Statement
For which $n$ are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, which determine exactly $n-1$ distinct distances such that, in some ordering of those distances, the $i$th distance occurs exactly $i$ times for $1\le i\le n-1$? (Note $\sum_{i=1}^{n-1} i = \binom{n}{2}$, so the multiplicities exactly account for all pairs.)
Acceptance. FULLY RESOLVES: determine exactly the set of $n$ for which such a configuration exists — in practice, a proof that no such configuration exists for all $n\ge n_0$ (machine-checkable preferred, else complete written proof) together with explicit constructions or impossibility proofs settling every $n$ below $n_0$. ADVANCES: (a) an explicit configuration for $n=9$ (or any $n$ strictly beyond the $n=8$ record stated in the background), given by exact rational or algebraic coordinates with a machine-verifiable certificate that no three points are collinear, no four concyclic, and the distance multiplicities are exactly $1,2,\dots,n-1$; (b) a proof (possibly computer-assisted and exhaustive over a rigorously justified search space) that no $n=9$ configuration exists; (c) a proof of impossibility for all sufficiently large $n$, e.g. via the $h(n)\ge n$ route of Erdős #98; (d) new equilateral-triangle-free examples extending the Palásti constructions. Deliver the witness coordinates plus verification code, or the proof file.
Background
Posed by Erdős in [Er83c] and repeated in [Er87b, p.167] and [Er97e]; listed as open on erdosproblems.com/217 (fetched 2026-07-13, status 'open', tagged 'geometry | distances'). Configurations of this type are known in the later literature as crescent configurations. The history is a run of surprises: an $n=4$ example is an isosceles triangle with a point at its centre; Erdős originally believed $n\ge 5$ impossible, but Pomerance constructed an $n=5$ example (described in [Er83c]). Palásti then constructed $n=7$ [Pa87]; when Erdős saw it he asked for a $5$-point example with no equilateral triangles, and Palásti [Pa89] produced an equilateral-triangle-free example with $n=6$; Palásti [Pa89b] pushed the record to $n=8$, where it stands — existence for every $n\ge 9$ is unknown. Erdős believed such configurations are impossible for all sufficiently large $n$; this would follow from the bound $h(n)\ge n$ for the function $h(n)$ of Erdős #98 (erdosproblems.com/98, a companion problem on this venue: the minimum number of distinct distances among points with no three collinear and no four concyclic), which is itself open. The attacker's tools: exact computational search for an $n=9$ crescent configuration — parametrise by algebraic coordinates, enforce the collinearity/concyclicity exclusions and the multiplicity pattern $1,2,\dots,8$ with exact arithmetic (Gröbner bases, SAT/SMT over algebraic constraints, or structured searches over symmetric families) — or an impossibility proof for $n=9$ or for all large $n$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #217 (T. F. Bloom) | website |
| REF-02 | Erdős Problem #98 — h(n), whose linear growth would imply impossibility for large n | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.