On the Number of Linear Multipartite Hypergraphs with Given Size

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.

Most of this work was finished when Fang Tian was a visiting research fellow at Australian National University. She is very grateful for what she learned there. She was supported by NSFC (No. 11871377, 12071274). She is also immensely grateful to the anonymous reviewer for his/her detailed and helpful suggestions.

The work was partially supported by NSFC (No.11871377).

Authors and Affiliations

1. Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

   Fang Tian

Corresponding author

Correspondence to Fang Tian.

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