Forbidden subgraphs in divisor graphs and an Erd\H{o}s divisibility problem

Erd\H{o}s asked for the largest size $f(n)$ of a subset of $\{1,\dots,n\}$ with no element dividing two others. We show that $f(n)=c_2\,n+o(n)$ for an effectively computable constant $c_2$, and moreover that the number $q(n)$ of such subsets satisfies $q(n)=\beta_2^{n+o(n)}$ for a computable constant $\beta_2$. To prove this, we recast the divisibility constraint as forbidding a certain directed subgraph in the divisor graph on $\{1,\dots,n\}$ and prove a more general result: for any finite family of connected forbidden subgraphs of the divisor graph, both the extremal density and counting rate are effectively computable. The proof uses a theorem of McNew on local statistics of divisor graphs.