OPEN

Let $F(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain any set of the form $\{n,2n,3n\}$. What is
\[ \lim_{N\to \infty}\frac{F(N)}{N}?\]
Is this limit irrational?

This limit was proved to exist by Graham, Spencer, and Witsenhausen [GrSpWi77]. Similar questions can be asked for the density or upper density of infinite sets without such configurations.