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 $\left(\genfrac{}{}{0ex}{}{k}{3}\right)$ triples determine circles of different radii.Determine $n_k$.

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

To make (n_k) well-defined for (k\ge 4), “general position” has to include **no four concyclic** (and also no three collinear, so every triple really determines a circle): otherwise, you could take $n$ points on one circle and then *every* triple has the same circumradius, so no $k$-subset can have (\binom{k}{3}) disti…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.