Geodesic
Asymptotic Enumeration of Non-Uniform Linear Hypergraphs
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 1, pp. 219-230

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A linear hypergraph, also known as a partial Steiner system, is a collection of subsets of a set such that no two of the subsets have more than one element in common. Most studies of linear hypergraphs consider only the uniform case, in which all the subsets have the same size. In this paper we provide, for the first time, asymptotically precise estimates of the number of linear hypergraphs in the non-uniform case, as a function of the number of subsets of each size.
Keywords: Steiner system, linear hypergraph, asymptotic enumeration, switching method 
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Hasheminezhad, Mahdieh; McKay, Brendan D. Asymptotic Enumeration of Non-Uniform Linear Hypergraphs. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 1, pp. 219-230. http://geodesic.mathdoc.fr/item/DMGT_2022_42_1_a13/