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