Problems on the set of squarefree numbers

Let $N$ be a positive integer and $\operatorname{ES}_N$ the maximal cardinality of a subset $\mathscr A\subset\{1,\dots,N\}$ such that $a+a'$ is squarefree for all $a\in\mathscr A$, $a'\in\mathscr A$. For large $N$ we obtain new upper and lower bounds for $\operatorname{ES}_N$.