Problem 92

Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb{R}^2$ in which every $x\in A$ has at least $f(n)$ points in $A$ equidistant from $x$ .

Is it true that $f(n)\leq n^{o(1)}$ ? Or even $f(n) < n^{O(1/\log\log n)}$ ?

Source: erdosproblems.com/92

Geometry
Distances

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

Disproved. The site's export of 2026-09-04 labels the problem "DISPROVED" (page last edited 21 May 2026), and its remarks say that the disproof of Problem 90 disproves this stronger form; see the claim page (OpenAI, 2026).

ErFi97: Erdős, Paul and Fishburn, Peter, Minimum planar sets with maximum equidistance counts. Comput. Geom. (1997), 207-218.
JJMT24: B. Janzer, O. Janzer, A. Methuku, and G. Tardos, Tight bounds for intersection-reverse sequences, edge-ordered graphs and applications. arXiv:2411.07188 (2024).
PaSh92: Pach, János and Sharir, Micha, Repeated angles in the plane and related problems. J. Combin. Theory Ser. A (1992), 12-22.
OpenAI26: OpenAI, Planar Point Sets with Many Unit Distances. Unnumbered 18-page technical report (2026).
ABGLSSTWW26: N. Alon, T. F. Bloom, W. T. Gowers, D. Litt, W. Sawin, A. Shankar, J. Tsimerman, V. Wang, and M. Matchett Wood, Remarks on the disproof of the unit distance conjecture, arXiv:2605.20695v1 (2026).

Problem 92

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