Status
On this page
Status
Topics
Status
On this page
Status
Topics
What is the minimum number of circles determined by any points in , not all on a circle?
What is the minimum number of circles determined by any points in , not all on a circle or a line?
Source: erdosproblems.com/506
A full solution has been claimed but not yet accepted. Settled in another form, for example when its parts resolve differently or the question is open-ended.
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 the minimum is , so only the values for remain. Elliott's original bound 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 ; it is AI-assisted and carries no acceptance evidence.
Read as the site words it, the question is degenerate: collinear points are not all on a circle and determine no circle, so the minimum under the site's wording is . 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 for it, except at .