Erdős–Fuchs conjecture on additive representation functions

Let $\mathcal{A} = \{a_{1}, a_{2},\ldots\}$ be a set of nonnegative integers satisfying $a_n \leq cn^2$ for all $n$, where $c>0$ is a real constant. Let $R_A(n)$ denote the number of representations of $n$ as a sum $a_i+a_j$ with $a_i,a_j\in A$ and $i\leq j$. Erdős–Fuchs conjecture. One has

$$\limsup_{n\to\infty}R_A(n)=\infty.$$

This conjecture strengthens the Erdős–Turán conjecture, which asserts that an additive basis of order two cannot have a bounded representation function. Its status is not resolved in the supplied source.