$F(N)<3\sqrt{N}+1$ for non-dividing subsets of $\{1,\ldots,N\}$

For every $N$, the largest non-dividing subset of $\{1,\ldots,N\}$ satisfies

This explicit bound is due to Erdős, Lev, Rauzy, Sándor and Sárközy, who introduced the term non-dividing for the property. It is far stronger than the elementary bound $|A|\le\min A$, and is the best explicit constant in the literature; asymptotically it has since been superseded by $F(N)\le N^{1/4+o(1)}$, which follows from the theorem of Pham and Zakharov on non-averaging sets, since every non-dividing set is non-averaging.

Together with Csaba's construction giving $F(N)\gg N^{1/5}$, this brackets the extremal function between $N^{1/5}$ and $N^{1/4+o(1)}$. Determining the correct order of growth of $F(N)$ is open, and is what Erdős problem #131 asks for.