Maximum cliques of the transitive closure of a chordal DAG Let $G=(V,A)$ be a directed acyclic graph, for which the underlying
undirected graph is chordal (so that every induced cycle in the
underlying undirected graph is a triangle).
It is known that in a chordal graph the number of maximum cliques
is linearly bounded in the number $|V|$ of vertices.
Let $G'=(V,A')$ be the transitive closure of $G$, so that for
every directed path $p= (v_i,\ldots,v_j)$ from $v_i$ to $v_j$
in $G$, there exists a directed edge $(v_i,v_j)$ in $A'$.

Question: Is there any polynomial/linear bound on the number
of maximum cliques in the undirected graph that underlies $G'$?
I think you can extend Vinicius dos Santos' idea to show that no polynomial bound is possible.
Consider a graph on $n$ vertices divided into $d\geq 1$ groups of size about $n/d$ as follows:

Its transitive closure has about $(\frac{n}{d})^d$ maximal (undirected) cliques.