Problem 827

Erdős Problem #827: declared status 'open'. Formalized: no. 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 $n_k$. Current best: Martinez and Rold\'{a}n-Pensado give a corrected argument that proves $n_k\ll k^9$. Prize: no. Tags: geometry.