Problem 104

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Statement. Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.

Status. Open.

Source. erdosproblems.com/104, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #104, https://www.erdosproblems.com/104.

References.

• [El84] Elekes, G., $n$ points in the plane can determine $n^{3/2}$ unit circles. Combinatorica (1984), 131.
• [Er75h] Erdős, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.
• [Er81d] Erdős, P., Some applications of graph theory and combinatorial methods to number theory and geometry. Algebraic methods in graph theory, Vol. I, II (Szeged, 1978) (1981), 137-148.
• [Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.
• [HaMe86] Harborth, Heiko and Mengersen, Ingrid, Point sets with many unit circles. Discrete Math. (1986), 193-197.

Formalization. Statement in formal-conjectures.

Current assessment

This account rests on the site's page and on the library cards of the two release preprints named below; no dated status search beyond the site is recorded, no claim is recorded for this problem, and the standing is the site's label. In [Er81d] Erdős states, without giving an argument, that at most $n(n-1)$ unit circles pass through three or more of $n$ points (source page). A double count, each pair of points lying on at most two unit circles, gives at most $n(n-1)/3$, as Harborth and Mengersen [HaMe86] note, and that order is what the question asks to beat. Elekes [El84] gives $n$-point sets determining $\gg n^{3/2}$ such circles, as the site reports, so the answer, if yes, cannot be improved below that order; in [Er92e] Erdős offered a prize for a proof or disproof that the count is $O(n^{3/2})$.

One result of 2026 is recorded here because it concerns unit distances among planar points and could be mistaken for progress on this problem. OpenAI's preprint A power saving for planar unit distances (OpenAI Math Release, 23 September 2026, pinned PDF, carded at openai_2026_power_saving_planar_unit_distances) proves that for some absolute $\beta<4/3$ every $n$-point planar set has at most $O(n^\beta)$ unordered pairs at unit distance. That is a bound on unit distances, a statement about pairs of points; this problem asks about unit circles through at least three of the points. Neither that preprint nor the release's companion The weak pinned planar distance theorem, carded at openai_2026_weak_pinned_planar_distance_theorem, which concerns the distinct distances from a point, states or cites a bound on the number of unit circles through at least three of the points. The unit-distance preprint does use point–unit-circle incidence bounds (its Proposition 2.3 is Székely's bound weakened by a $\log^2$ factor), but only near the balanced scale, and at the three-rich end such bounds give nothing below $n^2$. So the release gets no claim page here and no consequence for this problem is derived from it.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1975_problems_elementary_geometry
• erdos_1975_problems_elementary_geometry / equation_1
• erdos_1975_problems_elementary_geometry / equation_2
• erdos_1981_applications_graph_theory_combinatorial_methods_number
• erdos_1981_applications_graph_theory_combinatorial_methods_number / unit_circles_p143
• openai_2026_power_saving_planar_unit_distances
• openai_2026_weak_pinned_planar_distance_theorem