Erdős Problem 438

How large can $A \subseteq \{1, \ldots, N\}$ be if $A + A$ contains no square numbers?

A problem of Erdős [Er80, Er80c, ErGr80]. Taking all integers $\equiv 1 \pmod 3$ gives $|A| \ge N/3$, and Massias observed that all integers $\equiv 1, 5, 9, 13, 14, 17, 21, 25, 26, 29, 30 \pmod{32}$ give $|A| \ge \frac{11}{32} N$. Lagarias, Odlyzko and Shearer [LOS83] proved that $11/32$ is sharp for the modular version of the problem, and Khalfalah, Lodha and Szemerédi [KLS02] proved that it is sharp in general: the maximal such $A$ satisfies $|A| \le (\frac{11}{32} + o(1)) N$.