Vertex Turán problems for the oriented hypercube

Abstract

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