---

Claim. George B. Purdy and Justin W. Smith, Lines, circles, planes and spheres, Discrete Comput. Geom. 44 (2010), no. 4, 860–882; posted as arXiv:0907.0724 on 3 July 2009; carded at purdy_2009_lines_circles_planes_spheres. Write $f(n)$ for the least number of circles determined by $n$ points of the plane that do not all lie on one circle or one line, a circle being determined when it passes through at least three of the points. Elliott's 1967 theorem (its claim page) asserted $f(n)\ge\binom{n-1}{2}$ for $n>393$. Section 2.1 of the paper observes that this is wrong: $n-1$ points on a circle and one point $p$ off it, placed so that $p$ lies on $\lfloor(n-1)/2\rfloor$ lines through two of the circle points, determine only

$$1+\binom{n-1}{2}-\Bigl\lfloor\frac{n-1}{2}\Bigr\rfloor$$

circles, and asserts that Elliott's proof can easily be modified to give exactly this lower bound for the same range $n\ge394$. The modified proof is not printed there, and none of the cited sources prints it: the value for every $n\ge394$ rests on Elliott's 1967 argument as Purdy and Smith say it can be modified, with the circle-and-point configuration attaining the bound. The paper records that Bálintová and Bálint had printed the corrected bound in 1994 without explanation (their claim page), which the authors and Elliott had taken for a misprint, and that Segre's eight-point counterexample to Elliott's bound, the projection of a cube, had not revealed the general construction. The same section proves a further bound, Corollary 2.6, for sets with at most $n-k$ points on any line or circle, which is not part of this claim.

Covers. The value of $f(n)$ for every $n\ge394$. What remains of Problem 506 is the finite list of values $f(n)$ for $4\le n\le393$, which is why the site labels the problem resolved up to a finite check. The result takes the nondegeneracy condition to be that the points are not all on one circle or one line; the site's statement says only that they are not all on a circle, and its remark that some such condition is intended is recorded on the problem page. Wrona's claim asserts the remaining values, with the same formula from $n=9$ on, and is pending.

Depends on. No page of this wiki.

Acceptance. The result is refereed: Discrete and Computational Geometry published the paper, whose Section 2.1 asserts the modification of Elliott's argument without printing it, so the refereed text vouches for the counterexample and for that assertion. The site's curator, Thomas Bloom, labels the problem DECIDABLE and credits Purdy and Smith with observing the error in Elliott's proof and the corrected bound, best possible for all $n>393$ (problem page last edited 1 February 2026); the label DECIDABLE does not settle the problem, so the curator's credit is context and not acceptance evidence.