Erdős Problem 657

### What is the conjecture

https://www.erdosproblems.com/657

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 $f(n)\to \infty$?

Status: open

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

Read the conjecture at https://www.erdosproblems.com/657, then inspect existing formalized conjectures in the formal-conjectures repository to understand its Lean conventions. The work is complete when Problem 657 has been added as a formal Lean statement consistent with the repository’s existing structure.

Written by the indexing model from the issue text.