Problem 98

Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$ , with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\to \infty$ ?

Source: erdosproblems.com/98

Geometry
Distances

Wiki page

Status

Open

No claim settles this problem.

Open: the site labels the problem OPEN (snapshot of 5 September 2026). No result about the problem has been claimed, so it has no claim page and the frontmatter standing is open.

EFPR93: Erdős, Paul and Füredi, Zoltán and Pach, János and Ruzsa, Imre Z., The grid revisited. Discrete Math. (1993), 189--196.
Du08: A. Dumitrescu, On distinct distances among points in general position and other related problems. Period. Math. Hungar. 57 (2008), 165--176, DOI 10.1007/s10998-008-8165-4.
Ta24: T. Tao, Planar point sets with forbidden 4-point patterns and few distinct distances. arXiv:2409.01343v1; Discrete Comput. Geom. 76 (2026), 643--651, DOI 10.1007/s00454-025-00761-2.
GGK25: A. Ghosal, R. Goenka, and P. Keevash, On subsets of lattice cubes avoiding affine and spherical degeneracies. arXiv:2509.06935v1; Discrete Comput. Geom. 76 (2026), 1886--1911, DOI 10.1007/s00454-026-00853-7.

Problem 98

Status

Proof claims

Discussion