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 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 \(n\rightarrow \infty \) 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.

对于任何给定的整数\(r\geqslant 3\)，让\(k=k(n)\)是一个带有\(r\leqslant k\leqslant n\)的整数。如果每条边都是一组r个顶点，则超图是r均匀的，如果两条边最多在一个顶点上相交，则称该超图是线性的。令\(A_1,\ldots ,A_k\)成为[ n ]的给定k分区，其中\(|A_i|=n_i\geqslant 1\)。一个ř -uniform超图ħ称为ķ -三方如果每个边缘Ë满足\（|电子\帽A_I | \ leqslant 1 \）为\(1\leqslant i\leqslant k\)。在本文中，当边数为\(m(n)=o(n^{\frac{ )顶点上的线性k -partite r -uniform 超图的数量是渐近确定的4}{3}})\)。对于\(k=n\)，它是顶点集 [ n ] 上具有\(m=o(n^{ \frac{4}{3}})\)边的线性r -均匀超图的数量。