Problem 1109

Erdős Problem #1109: declared status 'open'. Formalized: no. Let $f(N)$ be the size of the largest subset $A\subseteq \{1,\ldots,N\}$ such that every $n\in A+A$ is squarefree. Estimate $f(N)$. In particular, is it true that $f(N)\leq N^{o(1)}$, or even $f(N) \leq (\log N)^{O(1)}$? Current best: First studied by Erd\H{o}s and S\'{a}rk\"{o}zy [ErSa87], who proved$$\log N \ll f(N) \ll N^{3/4}\log N,$$and guessed the lower bound is nearer the truth. Konyagin [Ko04] improved this to$$\log\log N(\log N)^2\ll f(N) \ll N^{11/15+o(1)}.$$The infinite analogue of this problem is [1103]. Prize: no. OEIS: A392164, A392165. Tags: number theory.