Problem 827

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claims/: The 4 claim pages of Problem 827, one per claimant's result; the problem's standing derives from them.

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Statement. Let $n_k$ be minimal such that if $n_k$ points in $\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all $\binom{k}{3}$ triples determine circles of different radii.

Determine $n_k$.

Formulation. The value of $n_k$ depends on what general position means. Erdős's 1975 statement [Er75h] defines it as no three points on a line and no four on a circle, and this page reads the problem that way. Martínez and Roldán-Pensado [MaRo15] work with the weaker condition that no four points lie on a line or a circle, which admits more point sets and so can only raise $n_k$. Their upper bounds hold under both readings; under the weaker one only $7\le n_4\le9$ is known.

Status. Open.

Source. erdosproblems.com/827, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #827, https://www.erdosproblems.com/827.

References.

• [Er75h] Erdős, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.
• [Er78c] Erdős, P., Some more problems on elementary geometry. Austral. Math. Soc. Gaz. (1978), 52-54.
• [Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.
• [MaRo15] Martínez, L. and Roldán-Pensado, E., Points defining triangles with distinct circumradii. Acta Math. Hungar. 145 (2015), no. 1, 136-141.

Formalization. None recorded for the problem. The Lean development of Kiichi's proof that $n_4=7$ is linked from its claim page; this corpus has not built it.

Current assessment

The question. Erdős asked in [Er75h] whether $n_k$ exists at all, and said he could not prove that it does. In [Er78c] he gave an argument for an explicit polynomial bound, which misses a case. The site labels the problem OPEN.

Claims. Martínez and Roldán-Pensado prove, in a refereed paper, that $n_k$ exists and $n_k=O(k^9)$, with $n_4\le9$ and $n_5\le37$; their Section 2 locates the gap in Erdős's 1978 argument. Martínez-Sandoval, Raggi and Roldán-Pensado derive $n_k=O(k^5/\log k)$ from a sunflower anti-Ramsey theorem in an arXiv manuscript of 2015. Two independent proofs that $n_4=7$ were posted on the site's thread in September 2026: a computer-assisted one by sallerk and a Lean-checked one by Kiichi, whose co-authors are instances of Claude. Neither has an outside review.

Results without a claim page. Two thread posts give asymptotic bounds without a manuscript, so they have no page. On 10 May 2026 FlaredRain posted a random-deletion proof of $n_k\le(3k)^5$, which counts the pairs of triples with equal circumradius; the post says an unnamed AI model found its core, and the site's commentary credits the $k^5$ bound to it. The manuscript of Martínez-Sandoval, Raggi and Roldán-Pensado already gives a sharper bound. On 16 June 2026 SamKorsky posted the lower bound $n_k\ge k^2\exp(-4\sqrt{\log2\log k}-O(\log\log k))$, from a generic planar projection of a grid lifted to a paraboloid: a set with all circumradii distinct contains no parallelogram, so its preimage has distinct differences. The post says GPT-5.5 was used to write it and to check its calculations. Erdős's 1978 argument has no page, since it is incorrect.

Remaining gaps. On the accepted record, $n_k$ exists and $n_k=O(k^9)$. With the claimed results, $n_k\ll k^5/\log k$ and $n_4=7$, and SamKorsky's post gives $n_k\ge k^{2-o(1)}$. The order of growth of $n_k$ and every value with $k\ge5$ are open.

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 / question_p3
• erdos_1978_more_problems_elementary_geometry
• erdos_1978_more_problems_elementary_geometry / inequality_1
• martinez_2015_points_defining_triangles_distinct_circumradii
• martinez_2015_points_defining_triangles_distinct_circumradii / lemma_4_1
• martinez_2015_points_defining_triangles_distinct_circumradii / theorem_1_1
• martinez_2015_points_defining_triangles_distinct_circumradii / theorem_1_2