  In this paper, we extend the chromatic symmetric function $X$ to a chromatic
$k$-multisymmetric function $X_k$, defined for graphs equipped with a partition
of their vertex set into $k$ parts. We demonstrate that this new function
retains the basic properties and basis expansions of $X$, and we give a method
for systematically deriving new linear relationships for $X$ from previous ones
by passing them through $X_k$.
  In particular, we show how to take advantage of homogeneous sets of $G$
(those $S \subseteq V(G)$ such that each vertex of $V(G) \backslash S$ is
either adjacent to all of $S$ or is nonadjacent to all of $S$) to relate the
chromatic symmetric function of $G$ to those of simpler graphs. Furthermore, we
show how extending this idea to homogeneous pairs $S_1 \sqcup S_2 \subseteq
V(G)$ generalizes the process used by Guay-Paquet to reduce the
Stanley-Stembridge conjecture to unit interval graphs.
