Problem 655

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?

Source: erdosproblems.com/655

Geometry
Distances

Wiki page

Status

ClaimedDisproved

A full solution has been claimed but not yet accepted. The statement is false.

OPEN, the site's label. The site's commentary records Zach Hunter's observation that equally spaced points on a circle disprove the statement as printed and presumes that a general-position hypothesis was intended, and the page's database box flags the original source as ambiguous. The derived standing departs from the label: it is claimed and disproved, because the page shows only the site's Statement and Hunter's regular polygon disproves it; that claim stays pending because it is unrefereed and the corpus has not built the Lean proof that the formal-conjectures catalog records.

Problem 655

Status

Proof claims

Discussion