Problem 959

Let $A\subset \mathbb{R}^2$ be a set of size $n$ and let $\{d_1,\ldots,d_k\}$ be the set of distinct distances determined by $A$ . Let $f(d)$ be the number of times the distance $d$ is determined, and suppose the $d_i$ are ordered such that

f(d_1)\geq f(d_2)\geq \cdots \geq f(d_k).

Estimate

\max (f(d_1)-f(d_2)),

where the maximum is taken over all $A$ of size $n$ .

Source: erdosproblems.com/959

Geometry
Distances

Wiki page

Status

Open

No claim settles this problem.

Open. The site's label was OPEN on 2026-10-06, and its page credits no result beyond the $n\log n$ bound of [CDL25], on the Clemen--Dumitrescu--Liu claim page (2025). The proof-claims tab carries two partial claims, neither adopted by the site: Colin Snyder's claim of 2026-07-15, a lower bound of order $n^{1+c/\log\log n}$ on the largest gap with a Lean 4 archive, on the Snyder claim page (2026), and Theofil Xeff's claim of 2026-07-21, a lower bound of order $n^{1+c}$ for an absolute $c>0$ , on the Xeff claim page (2026).

Problem 959

Status

Proof claims

Discussion