Distinct distances under a no-three-concyclic-per-centre condition: at least $(1+c)n/2$? (Erdős #655)
Statement
Let $x_1,\ldots,x_n\in\mathbb{R}^2$ satisfy the condition that no circle whose centre is one of the points contains three of the other points. Must the $x_i$ then determine at least $$(1+c)\frac{n}{2}$$ distinct distances (counting distinct values of $\lVert x_i-x_j\rVert$), for some absolute constant $c>0$ and all sufficiently large $n$? Caveat: the literal conjecture is false — $n$ points equally spaced on a circle satisfy the hypothesis yet determine only $\lfloor n/2\rfloor$ distinct distances (observed by Zach Hunter) — so the intended question is understood to add a general-position hypothesis, e.g. that no three of the points are collinear and no four are concyclic.
Acceptance. FULLY RESOLVES: under the intended general-position hypothesis (no three collinear, no four concyclic), a proof that the number of distinct distances is at least $(1+c)\frac{n}{2}$ for some explicit $c>0$ and all large $n$; OR a proof that even under general position the bound $(1+c)\frac{n}{2}$ fails (a construction achieving $(\tfrac12+o(1))n$ distinct distances). Machine-checkable proof preferred, otherwise a complete written proof. ADVANCES, any of, each with a complete proof or a certified construction: (a) any distinct-distance lower bound strictly above the trivial $\frac{n-1}{2}$ for the constrained point sets — e.g. $(\tfrac12+c')n$ with an explicit $c'>0$; (b) a general-position construction achieving few distinct distances that pins the extremal constant from above; (c) a rigorous determination of the correct general-position hypothesis under which the conjecture is intended, with proof or counterexample. Deliver the proof, or the certified extremal/near-extremal construction.
Background
A problem of Erdős and Pach [Er97e]. The hypothesis easily implies that every single point determines at least $\frac{n-1}{2}$ distinct distances (if some point saw fewer, a circle centred there would contain three others), so the whole set determines at least $\frac{n-1}{2}$ distinct distances; the conjecture asks for the strictly larger $(1+c)\frac{n}{2}$. Zach Hunter's regular-$n$-gon example disproves the statement as literally written, and Bloom notes that — in the spirit of related Erdős conjectures — a general-position assumption (no three on a line, no four on a circle) was presumably intended. Listed as open on erdosproblems.com/655 (fetched 2026-07-21, status 'open', tagged 'geometry | distances'; the site marks uncaptured comment activity as 'partial'). Venue neighbours: the general-position distinct-distances problem (no three points on a line and no four on a circle — is the number of distinct distances superlinear?) is Erdős #98, closely related to the intended general-position reading here; Szemerédi's conjecture that $n$ points with no three collinear determine at least $\frac{n}{2}$ distinct distances (Erdős #1082) is the $\frac{n}{2}$ baseline that the $(1+c)\frac{n}{2}$ target here aims to strictly beat; and the pinned-distance variant assuming no four points on a circle is Erdős #654. Attacker's tool: distinct-distance incidence machinery (crossing-number / Guth–Katz-type estimates) specialised to the no-three-concyclic-per-centre restriction, plus computational search over small general-position configurations to look for near-extremal sets or to refute candidate lower bounds.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #655 (T. F. Bloom) | website |
| REF-02 | Erdős Problem #654 — pinned distances with no four points on a circle (venue neighbour) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.