A canonical version of Ramsey’s Theorem proved by Erdös and Rado, implies that given any acyclic digraph D, there exists a least integer ρc(D) = n, such that every arc colouring (with an arbitrary number of colours) of the transitive tournament TTncontains a canonically coloured D(in the sense of Erdös-Rado). It follows that if Pmis a directed path and Dis an acyclic digraph, then there exists a least integer ρ*(Pm, D) = nsuch that every arc coloring of TTn, with an arbitrary number of colours, contains either a Pmwith no two arcs of the same colour or a monochromatic D. Recently, Lefmann, Rödl and Thomas [4], Lefmann and Rödl [5] have studied the numbers ρ*(Pn, Pm) and ρ*(Pn, TTm). In this paper we find ρ*(Pn, Sm), where Smis the out-star and give bounds for ρc(Sm, n) where Sm, nis the directed star with min-arcs and nout-arcs at the centre.