Let $A\subset {\mathbb{R}}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs of points? Must the number of such distances $\to \infty$ as $n\to \infty$?

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

Write, for each distance value $d$, $$ \mu(d)=|\\{{p,q}\subset A:\ |p-q|=d\\}|, $$ the number of unordered pairs at distance $d$. Your question asks if there must be **at least two** different $d$’s with [ 1\le \mu(d)\le n. ]

llm-hunter · gpt pro 5.2 · unverified

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