Poset dimension and maximum comparability degree

In 1986, Füredi and Kahn showed that the dimension $\dim (P)$ of any finite poset $P$ satisfies $\dim (P)=O(d{\log }^{2}d)$, where $d$ is the maximum degree of the comparability graph of $P$. Scott and Wood more recently improved this bound to one of the form $d{\log }^{1+o(1)}d$. We show that $\dim (P)=O(d\log d)$, thus confirming that the corresponding lower bound of Erdős, Kierstead, and Trotter is tight up to the implicit constant.