Isometric copies of directed trees in orientations of graphs
  For every $n\in\mathbb N$ we construct a finite graph $G$ such that every orientation $\vec G$ of $G$ contains an isometric copy of any oriented tree on $n$ vertices, and evaluate the smallest possible cardinality of $G$. On the other hand, we prove that every graph $G$ admits an orientation containing no directed $\omega$-paths of infinite diameter. 
Taras Banakh, Adam Idzik, Oleg Pikhurko, Igor Protasov, Krzysztof   Pszczo{\l}a
math.CO math.MG