  We consider the geometry of random interlacements on the $d$-dimensional
lattice. We use ideas from stochastic dimension theory developed in
\cite{benjamini2004geometry} to prove the following: Given that two vertices
$x,y$ belong to the interlacement set, it is possible to find a path between
$x$ and $y$ contained in the trace left by at most $\lceil d/2 \rceil$
trajectories from the underlying Poisson point process. Moreover, this result
is sharp in the sense that there are pairs of points in the interlacement set
which cannot be connected by a path using the traces of at most $\lceil d/2
\rceil-1$ trajectories.
