  We prove the well-known Brown-Erd\H{o}s-S\'os Conjecture for hypergraphs of
large uniformity in the following form: any dense linear $r$-graph $G$ has $k$
edges spanning at most $(r-2)k+3$ vertices, provided the uniformity $r$ of $G$
is large enough given the linear density of $G$, and the number of vertices of
$G$ is large enough given $r$ and $k$.
