  First-Fit is a greedy algorithm for partitioning the elements of a poset into
chains. Let $\textrm{FF}(w,Q)$ be the maximum number of chains that First-Fit
uses on a $Q$-free poset of width $w$. A result due to Bosek, Krawczyk, and
Matecki states that $\textrm{FF}(w,Q)$ is finite when $Q$ has width at most
$2$. We describe a family of posets $\mathcal{Q}$ and show that the following
dichotomy holds: if $Q\in\mathcal{Q}$, then $\textrm{FF}(w,Q) \le 2^{c(\log
w)^2}$ for some constant $c$ depending only on $Q$, and if
$Q\not\in\mathcal{Q}$, then $\textrm{FF}(w,Q) \ge 2^w - 1$.
