Suppose we have a directed complete graph. Can we always find a subset $S\neq \emptyset$ of the vertices such that for every vertex $v$, $v$ has incoming edge from at least $|S|/2$ of the vertices in $S$? Note that $v$ can be in $S$. Also, we assume that each vertex has an incoming edge from itself.

Suppose we have a directed complete graph. Can we always find a subset $S\neq \emptyset$ of the vertices such that for every vertex $v$, $v$ has incoming edge from at least $\dfrac{|S|}{2}$ of the vertices in $S$? 

Note that $v$ can be in $S$. Also, we assume that each vertex has an incoming edge from itself.

Suppose we have a complete directed graph. Can we always find a subset $S$ of the vertices such that for every vertex $v$, $v$ has incoming edge from at least $|S|/2$ of the vertices in $S$? Note that $v$ can be in $S$. Also, we assume that each vertex has an incoming edge from itself.

Suppose we have a directed complete graph. Can we always find a subset $S\neq \emptyset$ of the vertices such that for every vertex $v$, $v$ has incoming edge from at least $|S|/2$ of the vertices in $S$? Note that $v$ can be in $S$. Also, we assume that each vertex has an incoming edge from itself.