  A path in the hypercube $Q_n$ is said to be a geodesic if no two of its edges
are in the same direction. Let $G$ be a subgraph of $Q_n$ with average degree
$d$. How long a geodesic must $G$ contain? We show that $G$ must contain a
geodesic of length $d$. This result, which is best possible, strengthens a
theorem of Feder and Subi. It is also related to the `antipodal colourings'
conjecture of Norine.
