Dense sets of natural numbers with unusually large least common multiples

For any constant $C_0>0$, we construct a set $A\subset \mathbb{N}$ such that one has $$\sum _{n\in A:n\le x}\frac{1}{n}=\exp ((\frac{C_0}{2}+o(1))(\log \log x{)}^{1\mathrm{/}2}\log \log \log x)$$ and $$\sum _{n,m\in A:n,m\le x}\frac{1}{\mathrm{lcm}(n,m)}{\ll }_{C_0}{\left(\sum _{n\in A:n\le x}\frac{1}{n}\right)}^2$$ as $x\to \mathrm{\infty }$, with the growth rate given here optimal up to the dependence on $C_0$. This answers in the negative a question of Erdős and Graham, and also clarifies the nature of certain ``mostly coprime’’ sets studied by Bergelson and Richter.