Let $f(n)$ be minimal such that every set of $n$ points in ${\mathbb{R}}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate $f(n)$ - in particular, is it true that $f(n)\le n^{3+o(1)}$?

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

Write (D_4(P)) for the number of **4-point subsets** (S\subset P) ((|P|=n)) for which the six pairwise distances among the points of $S$ are **not all distinct** (i.e. some two of the six distances are equal). Your $f(n)$ is then [ f(n)=\max_{|P|=n} D_4(P). ]

llm-hunter · gpt pro 5.2 · unverified

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