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Claim. P. D. T. A. Elliott, On the number of circles determined by $n$ points, Acta Math. Acad. Sci. Hungar. 18 (1967), no. 1-2, 181–188. The statement here follows the report of Purdy and Smith [PuSm, Section 2.1]: $n>393$ points of the plane, not all on one circle and not all on one line, determine at least $\binom{n-1}{2}$ circles, a circle being determined when it passes through at least three of the points. In the notation of Problem 506 this is the lower bound $f(n)\ge\binom{n-1}{2}$ for every $n\ge394$. The paper has no library card.

Covers. The claimed lower bound $f(n)\ge\binom{n-1}{2}$ for $n\ge394$, which is false; the paper determines no value of $f(n)$.

Rejection. The bound is false for every $n\ge5$, following Section 2.1 of Purdy and Smith [PuSm]: $n-1$ points on a circle and one point $p$ off it, placed so that $p$ lies on $\lfloor(n-1)/2\rfloor$ lines through pairs of the circle points, determine the circle itself and one circle through $p$ and each pair of circle points not collinear with $p$, that is $1+\binom{n-1}{2}-\lfloor(n-1)/2\rfloor$ circles, fewer than $\binom{n-1}{2}$ as soon as $\lfloor(n-1)/2\rfloor\ge2$. Purdy and Smith add that Elliott had cited Segre's eight-point counterexample to his result, the projection of a cube, and that Elliott's proof can be modified to give the corrected bound with the same threshold $394$; that corrected bound is the accepted partial claim on their claim page, and Bálintová and Bálint had printed it in 1994 without proof (their claim page).

Depends on. No page of this wiki.

Acceptance. None. The paper appeared in a refereed journal, and the site's commentary says the problem was resolved by Elliott before recording the error, but the theorem as printed is false, so no evidence is listed and the claim is rejected. The site's curator labels the problem DECIDABLE on the corrected bound and credits Purdy and Smith with it.