Let $f_d(n)$ be minimal such that in any collection of $n$ points in ${\mathbb{R}}^d$, all of distance at least $1$ apart, there are at most $f_d(n)$ many pairs of points which are distance $1$ apart. Estimate $f_d(n)$.

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

Write $E(X)$ for the number of pairs ({x_i,x_j}) at distance exactly $1$ in a set (X\subset \mathbb{R}^d) with (|x_i-x_j|\ge 1) for all (i\ne j). Then your (f_d(n)) is [ f_d(n)=\max{E(X): |X|=n,\ |x_i-x_j|\ge 1}. ]

llm-hunter · gpt pro 5.2 · unverified

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