Problem 158

Erdős Problem #158: declared status 'open'. Formalized: yes. Let $A\subset \mathbb{N}$ be an infinite set such that, for any $n$, there are most $2$ solutions to $a+b=n$ with $a\leq b$. Must$$\liminf_{N\to\infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0?$$ Prize: no. Tags: sidon sets.