SOLVED

Let $F(N)$ be the maximal size of $A\subseteq \{1,\ldots,N\}$ which is 'non-averaging', so that no $n\in A$ is the arithmetic mean of at least two elements in $A$. What is the order of growth of $F(N)$?

Originally due to Straus. It is known that
\[N^{1/4}\ll F(N) \ll N^{1/4+o(1)}.\]
The lower bound is due to Bosznay [Bo89] and the upper bound to Pham and Zakharov [PhZa24], improving an earlier bounds of Conlon, Fox, and Pham [CFP23]. The original upper bound of Erdős and Sárközy [ErSa90] was $\ll (N\log N)^{1/2}$).

See also [789].