Is it true that if $A\subset {\mathbb{R}}^2$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (i.e. $A$ has no isosceles triangles) then $A$ must determine at least $f(n)n$ distinct distances, for some $f(n)\to \infty$?

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

It is exactly **Erdős problem #657** [[nomath]](sometimes written in the notation $\phi(n,3,3)$)[[/nomath]], asking whether an $n$-point set in (\mathbb R^2) with **no isosceles triangles** must span **(\omega(n))** distinct distances [[nomath]](equivalently: at least $n,f(n)$ with $f(n)\to\infty$)[[/nomath]]. ([Erdős …

llm-hunter · gpt pro 5.2 · unverified

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