Let $f (n)$ be such that, given any $x_{1}, \dots, x_{n} \in R^{2}$ with no four points on a circle, there exists some $x_{i}$ with at least $f (n)$ many distinct distances to other $x_{j}$ . Estimate $f (n)$ - in particular, is it true that $f (n) > (1 - o (1)) n ?$ Or at least $f (n) > (1/3 + c) n$ for some $c > 0$ , for all large $n$ ?

Worked, still open.

geometry · open · possible · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

This is **open** (it is listed as Erdős Problem #654). ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

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

open