Problem 662

Consider the triangular lattice with minimal distance between two points $1$ . Denote by $f(t)$ the number of distances from any points $\leq t$ . For example $f(1)=6$ , $f(\sqrt{3})=12$ , and $f(3)=18$ .

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that $d(x_i,x_j)\geq 1$ for all $i\neq j$ . Is it true that, provided $n$ is sufficiently large depending on $t$ , the number of distances $d(x_i,x_j)\leq t$ is less than or equal to $f(t)$ with equality perhaps only for the triangular lattice?

In particular, is it true that the number of distances $\leq \sqrt{3}-\epsilon$ is less than $1$ ?

Source: erdosproblems.com/662

Geometry
Distances

Wiki page

Status

ClaimedDisproved

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

The site labels the problem OPEN and credits no result; its commentary says the wording does not make sense as written and fixes no intended form, so the label describes a question the site does not state. The wording fails under each counting reading (Formulation). Its proof-claims tab carries one claim, submitted 2026-07-15, which the Snyder claim page (2026) records, and its discussion thread carries Przemek Chojecki's note of 2026-04-23, which the Chojecki claim page (2026) records; each refutes the wording, the site adopts neither, and the frontmatter standing derives from those pending full claims. The page therefore departs from the site's OPEN label, a departure the Formulation explains: following the label would state as open a wording refuted under every counting reading.

Problem 662

Status

Proof claims

Discussion