Problem 506

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claims/: The 4 claim pages of Problem 506, one per claimant's result; the problem's standing derives from them.

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Statement. What is the minimum number of circles determined by any $n$ points in $\mathbb{R}^2$, not all on a circle?

Statement (corrected). What is the minimum number of circles determined by any $n$ points in $\mathbb{R}^2$, not all on a circle or a line?

Notes. Read as the site words it, the question is degenerate: $n\ge3$ collinear points are not all on a circle and determine no circle, so the minimum under the site's wording is $0$. Erdős's statement [Er61, p. 245] has the same wording, and the site remarks that some nondegeneracy condition is intended, either that the points are not all on one line or the stronger one that no three are collinear. The corrected Statement adds "or a line" and nothing else; it is the condition of Elliott [El67], Purdy and Smith [PuSm], Section 2.1, the formal-conjectures statement and Wrona's claim. Under the condition that no three points are collinear the question is a different one, with a different claimed answer: Wrona's repository claims $1+\binom{n-1}{2}$ for it, except $20$ at $n=8$.

Status. The site labels the problem DECIDABLE, a label it glosses as resolved up to a finite check (page last edited 1 February 2026; the label and the proof-claims thread as of 6 October 2026). The standing in the frontmatter derives from the claim pages. The accepted partial result is the corrected Elliott bound of Purdy and Smith: for $n\ge394$ the minimum is $1+\binom{n-1}{2}-\lfloor(n-1)/2\rfloor$, so only the values for $4\le n\le393$ remain. Elliott's original bound $\binom{n-1}{2}$ is the rejected claim Elliott 1967, and Bálintová and Bálint's bounds on circles through exactly three of the points are the accepted partial claim Bálintová and Bálint 1994. The pending full claim is Wrona's, posted on the site's proof-claims thread on 20 August 2026 with a manuscript and a Lean development, asserting the exact minimum for every $n\ge4$; it is AI-assisted and carries no acceptance evidence.

Source. erdosproblems.com/506, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #506, https://www.erdosproblems.com/506.

References.

• [BaBa94] Bálintová, A. and Bálint, V., On the number of circles determined by $n$ points in the Euclidean plane. Acta Math. Hungar. 63 (1994), no. 3, 283-289.
• [El67] Elliott, P. D. T. A., On the number of circles determined by $n$ points. Acta Math. Acad. Sci. Hungar. 18 (1967), no. 1-2, 181-188.
• [Er61] Erdős, P., Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221-254.
• [PuSm] Purdy, G. B. and Smith, J. W., Lines, circles, planes and spheres. arXiv:0907.0724 (2009); Discrete Comput. Geom. 44 (2010), no. 4, 860-882, doi:10.1007/s00454-010-9270-3.

Formalization. Statement in formal-conjectures (at its commit of 2026-10-07), with the hypothesis that the points are neither collinear nor concyclic; the file marks the question open, carries the value for $n>393$ and Segre's eight-point observation as solved variants, and attaches no formal proof. Wrona's repository states its answer in the same terms as an IsLeast theorem for every $n\ge4$; it is a link on the claim page and not verification evidence, which requires Lean this corpus built and audited.

Current assessment

The site's formulation (page last edited 1 February 2026) is degenerate as worded, and this page answers the corrected Statement: write $f(n)$ for the least number of circles through at least three of $n$ points of the plane that are not all on one circle or one line. Elliott [El67] asserted $f(n)\ge\binom{n-1}{2}$ for $n>393$, but the bound is wrong: $n-1$ concyclic points and one point off the circle lying on $\lfloor(n-1)/2\rfloor$ of their connecting lines determine only $1+\binom{n-1}{2}-\lfloor(n-1)/2\rfloor$ circles, and Segre's projection of a cube already gives fewer than $\binom{7}{2}$ circles for $n=8$. Purdy and Smith [PuSm], Section 2.1, record the counterexample and assert that Elliott's proof can be modified to give the corrected bound for the same range, so that the exact value of $f(n)$ for every $n\ge394$ rests on Elliott's 1967 argument as Purdy and Smith say it can be modified; none of the cited sources prints the modified proof, and Bálintová and Bálint [BaBa94] had printed the corrected bound in 1994 without explanation. That reduction to finitely many cases is the accepted partial claim. Apart from the elementary $f(4)=3$, given by three collinear points and one more, the values $f(n)$ for $5\le n\le393$ are open in the published literature as recorded here. Wrona's 2026 claim asserts $f(4),\dots,f(8)=3,5,8,11,17$ and the corrected Elliott formula for every $n\ge9$, under the hypothesis that the points are neither collinear nor concyclic; its manuscript and Lean development carry no acceptance evidence. The site's discussion thread carries two unpublished posts on small $n$, recorded here without claim pages because they are thread posts. On 11 June 2026 Yuriy Peysakhov posted the eight points $(\pm1,\pm1)$ and $(\pm2,\pm2)$, found with Citadel, an AI-assisted verified-discovery engine he is building; they determine $18$ circles, so $f(8)\le18<19$. On 18 August 2026, two days before Wrona's claim, the user mzn posted, with AI assistance, the values $f(5),\dots,f(8)=5,8,11,17$ with explicit configurations. The lower bounds at $n=6$ and $n=8$ come from an exhaustive enumeration of the possible systems of lines and circles, and the bound at $n=7$ from that enumeration together with a short synthetic argument that rules out eight circles. These values agree with Wrona's, and Peysakhov's bound is consistent with them. Elliott's false bound is the rejected claim on its claim page, and Bálintová and Bálint's bounds on three-point circles, $k_3\ge5n(n-1)/133$ for every $n\ge4$, are the accepted partial claim on theirs. No independent review of any result on this page is recorded. The search scope is the site's page export of 2026-09-04, the site's proof-claims thread as of 6 October 2026, its discussion thread as of 7 October 2026, and the formal-conjectures statement file at its commit of 2026-10-07.

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.

• purdy_2009_lines_circles_planes_spheres
• purdy_2009_lines_circles_planes_spheres / corollary_2_6
• purdy_2009_lines_circles_planes_spheres / remark_p8