An antichain in a DAG $(V, E)$ is a subset $A \subseteq V$ of vertices that are pairwise unreachable, namely, there are no $v \neq v' \in A$ such that $v$ is reachable from $v'$ in $E$. From Dilworth's theorem in partial order theory, it is known that if DAG has no antichain of size $k \in \mathbb{N}$, then it can be decomposed in a union of at most $k-1$ disjoint chains, i.e., directed paths.
Now, I am interested in labeled DAGs, i.e., DAGs where each vertex $v$ carries a label $\lambda(v)$ in some fixed finite set $\Sigma$ of labels. Given an antichain $A \subseteq V$, I can define its labeled size as the minimal number of occurrences of the labels of $\Sigma$ in $A$, namely, $\min_{a \in \Sigma} |\{v \in A \mid \lambda(v) = a\}|$. Is there an analogue of Dilworth's theorem in this context? In other words, if I assume that a DAG has no antichain of labeled size $k \in \mathbb{N}$, what can I assume about its structure? Can I decompose it in some special way? I am already puzzled by the case of $\Sigma = \{a, b\}$, but also interested in the case of a general finite label set.
To visualize this for $\Sigma = \{a, b\}$, saying that $G$ has no antichain of labeled size $k$ means that there is no antichain containing at least $k$ vertices labeled $a$ and $k$ vertices labeled $b$; there can be arbitrarily large antichains but they have to contain only $a$ elements or only $b$ elements, up to $k-1$ exceptions at most. It seems that disallowing large antichains should enforce that the DAG essentially "alternates" between parts of large width for $a$-labeled vertices, and large width for $b$-labeled vertices, but I have not been able to formalize this intuition. (Of course, a suitable structural characterization must talk about the labels of vertices in addition to the shape of the DAG, because already for $k \geq 1$ and on $\{a, b\}$ the condition is satisfied by completely arbitrary DAGs whenever all vertices carry the same label.)