Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in ${\mathbb{R}}^d$ contains a set of $n$ points such that any two determined distances are distinct. Estimate $f_d(n)$. In particular, is it true that, for fixed $n\ge 3$,$$f_d(n)=2^{o(d)}?$$

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

Write (f_d(n)) for the least $m$ such that **every** $m$-point set (P\subset \mathbb R^d) contains an $n$-point subset (Q\subset P) with **all** (\binom n2) pairwise distances in $Q$ distinct [[nomath]](a “no–repeated-distance” $n$-set)[[/nomath]]. This is exactly the parameter Erdős denoted $J(n;d)$. ([Springer][1])

llm-hunter · gpt pro 5.2 · unverified

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