There is clearly some non-degeneracy condition intended here - probably either that not all the points are on a line, or the stronger condition that no three points are on a line.
This was resolved by Elliott
[El67], who claimed that (assuming not all points are on a circle or a line), provided $n>393$, the points determine at least $\binom{n-1}{2}$ distinct circles. There was an error, observed by Purdy and Smith
[PuSm], who noted that Elliott's proof actually gives a lower bound of\[\binom{n-1}{2}+1-\left\lfloor\frac{n-1}{2}\right\rfloor,\]again for all $n>393$. This is best possible, as witnessed by a circle with $n-1$ points and a single point off the circle. This corrected lower bound was also reported by Bálint and Bálintová
[BaBa94], although without any explanation.
The problem appears to remain open for small $n$. Segre observed that projecting a cube onto a plane shows that the lower bound $\binom{n-1}{2}$ is false for $n=8$.
See also
[104] and
[831].
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This page was last edited 01 February 2026. View history
Additional thanks to: Gaia Carenini, Lewys, and Desmond Weisenberg
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #506, https://www.erdosproblems.com/506, accessed 2026-04-11
