Independent Chains in Acyclic Posets

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 $a(\mathcal{G})=(n+o(n))\mathrm{/}{\log }_2(n)$ in this class.