---

Claim. A. Bálintová and V. Bálint, On the number of circles determined by $n$ points in the Euclidean plane, Acta Math. Hungar. 63 (1994), no. 3, 283–289; the theorems are stated as its zbMATH review (Zbl 0796.51008) gives them. Let $P$ be a set of $n\ge4$ points of the plane, not all on one circle or one line. Theorem 1 states that every point $p$ of $P$ lies on at least $15(n-1)/133$ circles containing exactly three points of $P$; Theorem 2 states that the number $k_3$ of circles containing exactly three points of $P$ satisfies $k_3\ge5n(n-1)/133$. Every such circle is one of the circles determined by $P$, so in the notation of Problem 506 the paper gives $f(n)\ge5n(n-1)/133$ for every $n\ge4$ (a remark of this page). The paper also prints the lower bound $1+\binom{n-1}{2}-\lfloor(n-1)/2\rfloor$ for $n>393$, the corrected form of Elliott's theorem, without proof; Purdy and Smith record that they, and Elliott, had taken it for a misprint. That bound is the accepted partial claim on Purdy and Smith's page, and Elliott's original bound is the rejected claim on its own page. The paper has no library card.

Covers. The lower bounds of Theorems 1 and 2 on circles through exactly three of the points, for every $n\ge4$, and the lower bound $f(n)\ge5n(n-1)/133$ they imply. The paper determines no value of $f(n)$; the corrected Elliott bound it prints is not proved there.

Depends on. No page of this wiki.

Acceptance. Refereed: Acta Mathematica Hungarica published the paper. The site's curator labels the problem DECIDABLE and credits Purdy and Smith; the remark that this paper reported the corrected bound without explanation is context and not acceptance evidence.