On the Number of Linear Multipartite Hypergraphs with Given Size

For any given integer r ⩾ 3 , let k = k ( n ) be an integer with r ⩽ k ⩽ 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 , … , A k be a given k -partition of [ n ] with | A i | = n i ⩾ 1 . An r -uniform hypergraph H is called k - partite if each edge e satisfies | e ∩ A i | ⩽ 1 for 1 ⩽ i ⩽ k . In this paper, the number of linear k -partite r -uniform hypergraphs on n → ∞ vertices is determined asymptotically when the number of edges is m ( n ) = o ( n 4 3 ) . For k = n , it is the number of linear r -uniform hypergraphs on vertex set [ n ] with m = o ( n 4 3 ) edges.