Erdős Problem 819

### What is the conjecture

https://www.erdosproblems.com/819

Let $f(N)$ be maximal such that there exists $A\subseteq \\{1,\ldots,N\\}$ with $\lvert A\rvert=\lfloor N^{1/2}\rfloor$ such that $\lvert (A+A)\cap [1,N]\rvert=f(N)$. Estimate $f(N)$.

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

Start from the conjecture statement at erdosproblems.com/819 and review existing formalized conjectures in the formal-conjectures repository. Determine how this open problem should be represented in Lean and what verification would establish a complete addition.

Written by the indexing model from the issue text.