Problem 168

Erdős Problem #168: declared status 'open'. Formalized: yes. 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? Prize: no. OEIS: A004059, A057561, A094708, A386439. Tags: additive combinatorics.