Vertex Turán problems for the oriented hypercube

In this short note we consider the oriented vertex Turán problem in the hypercube: for a fixed oriented graph F→ \vec F , determine the maximum cardinality exv(F→,Q→n) e{x_v}\left( {\vec F,{{\vec Q}_n}} \right) of a subset U of the vertices of the oriented hypercube Q→n {\vec Q_n} such that the induced subgraph Q→n[ U ] {\vec Q_n}\left[ U \right] does not contain any copy of F→ \vec F . We obtain the exact value of exv(Pk,→ Qn→) e{x_v}\left( {\overrightarrow {{P_k},} \,\overrightarrow {{Q_n}} } \right) for the directed path Pk→ \overrightarrow {{P_k}} , the exact value of exv(V2→, Qn→) e{x_v}\left( {\overrightarrow {{V_2}} ,\,\overrightarrow {{Q_n}} } \right) for the directed cherry V2→ \overrightarrow {{V_2}} and the asymptotic value of exv(T→,Qn→) e{x_v}\left( {\overrightarrow T ,\overrightarrow {{Q_n}} } \right) for any directed tree T→ \vec T .