  We consider the problem of determining the maximum order of an induced
vertex-disjoint union of cliques in a graph. More specifically, given some
family of graphs $\mathcal{G}$ of equal order, we are interested in the
parameter $a(\mathcal{G}) = \min_{G \in \mathcal{G}} \max \{ |U| : U \subseteq
V, G[U] \text{ is a vertex-disjoint union of cliques} \}$. We determine the
value of this parameter precisely when $\mathcal{G}$ is the family of
comparability graphs of $n$-element posets with acyclic cover graph. In
particular, we show that $a(\mathcal{G}) = (n+o(n))/\log_2 (n)$ in this class.
