  In 2006, Noga Alon raised the following open problem: Does there exist an
absolute constant $c>0$ such that every $2n$-vertex digraph with minimum
out-degree at least $s$ contains an $n$-vertex subdigraph with minimum
out-degree at least $\frac{s}{2}-c$ ? In this note, we answer this natural
question in the negative, by showing that for arbitrarily large values of $n$
there exists a $2n$-vertex tournament with minimum out-degree $s=n-1$, in which
every $n$-vertex subdigraph contains a vertex of out-degree at most
$\frac{s}{2}-\left(\frac{1}{2}+o(1)\right)\log_3(s)$.
