  A family $\mathcal{P}$ of subgraphs of $G$ is called a {\it path cover}
(resp. a {\it path partition}) of $G$ if $\bigcup _{P\in \mathcal{P}}V(P)=V(G)$
(resp. $\dot\bigcup _{P\in \mathcal{P}}V(P)=V(G)$) and every element of
$\mathcal{P}$ is a path. The minimum cardinality of a path cover (resp. a path
partition) of $G$ is denoted by ${\rm pc}(G)$ (resp. ${\rm pp}(G)$). In this
paper, we characterize the forbidden subgraph conditions assuring us that ${\rm
pc}(G)$ (or ${\rm pp}(G)$) is bounded by a constant. Our main results introduce
a new Ramsey-type problem.
