  For any given integer $r\geqslant 3$, let $k=k(n)$ be an integer with
$r\leqslant k\leqslant n$. A hypergraph is $r$-uniform if each edge is a set of
$r$ vertices, and is said to be linear if two edges intersect in at most one
vertex. Let $A_1,\ldots,A_k$ be a given $k$-partition of $[n]$ with
$|A_i|=n_i\geqslant 1$. An $r$-uniform hypergraph $H$ is called {\it
$k$-partite} if each edge $e$ satisfies $|e\cap A_i|\leqslant 1$ for
$1\leqslant i\leqslant k$. In this paper, the number of linear $k$-partite
$r$-uniform hypergraphs on $n\to\infty$ vertices is determined asymptotically
when the number of edges is $m(n)=o(n^{\frac{4}{3}})$. For $k=n$, it is the
number of linear $r$-uniform hypergraphs on vertex set $[n]$ with $m=o(n^{
\frac{4}{3}})$ edges.
