Geodesic
The representation of multi-hypergraphs by set intersections
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 3, pp. 565-582

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This paper deals with weighted set systems (V,,q), where V is a set of indices, ⊂ 2^V and the weight q is a nonnegative integer function on . The basic idea of the paper is to apply weighted set systems to formulate restrictions on intersections. It is of interest to know whether a weighted set system can be represented by set intersections. An intersection representation of (V,,q) is defined to be an indexed family R = (R_v)_v∈ V of subsets of a set S such that
Keywords: intersection graph, intersection hypergraph 
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Bylka, Stanisław; Komar, Jan. The representation of multi-hypergraphs by set intersections. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 3, pp. 565-582. http://geodesic.mathdoc.fr/item/DMGT_2007_27_3_a14/