Problem Statement
Let $n_k$ be minimal such that if $n_k$ points in $\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all $\binom{k}{3}$ triples determine circles of different radii.
Determine $n_k$.
Determine $n_k$.
Categories:
Geometry
Progress
In [Er75h] Erdős asks whether $n_k$ exists. In [Er78c] he gave a simple argument which proves that it does, and in fact\[n_k \leq k+2\binom{k-1}{2}\binom{k-1}{3},\]but this argument is incorrect, as explained by Martinez and Roldán-Pensado [MaRo15].Martinez and Roldán-Pensado give a corrected argument that proves $n_k\ll k^9$.
Source: erdosproblems.com/827 | Last verified: January 16, 2026