Erdős problem 856

Let $k\geq 3$ and $f_k(N)$ be the maximum of $\sum_{n\in A}\frac{1}{n}$ over all $A\subseteq\{1,\ldots,N\}$ containing no $k$ subsets with the same pairwise least common multiple. Estimate $f_k(N)$. The claimed answer: $f_k(N)=(\log N)^{\gamma_k+o(1)}$, where $\gamma_k$ is a weighted generalization of the Tang-Zhang sunflower capacity.