The Second Term for Strongly 2-Primitive Sets

Let $F(n)$ be the largest size of a set $A\subseteq [1,n]$ such that $a\nmid bc$ whenever $a,b,c\in A$ and $a\mathrm{\notin }\{b,c\}$, with $b$ and $c$ allowed to coincide. We prove $$F(n)=\pi (n)+(\frac{27}{2}+o(1))\frac{n^{2\mathrm{/}3}}{(\log n{)}^2}\mathrm{.}$$ This determines the second-order constant conjectured by Erdős; the upper bound keeps the leading constants in his multiplicative basis, while the lower bound packs scale-separated prime triples by proper edge-colourings.