The representation of multi-hypergraphs by set intersections

This paper deals with weighted set systems (V, E , q), where V is a set of indices, E ⊂ 2 and the weight q is a nonnegative integer function on E . 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, E , q) is defined to be an indexed family R = (Rv)v∈V of subsets of a set S such that ∣∣ ⋂ v∈E Rv ∣∣ = q(E) for each E ∈ E . A necessary condition for the existence of such representation is the monotonicity of q on E i.e., if F ⊂ E then q(F ) ≥ q(E). Some sufficient conditions for weighted set systems representable by set intersections are given. Appropriate existence theorems are proved by construction of the solutions. 566 S. Bylka and J. Komar The notion of intersection multigraphs to intersection multihypergraphs — hypergraphs with multiple edges, is generalized. Some conditions for intersection multi-hypergraphs are formulated.