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This paper deals with weighted set systems (V,,q), where V is a set of indices, $\subset 2^V$ 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 $R={\left(R_v\right)}_{v\in V}$ of subsets of a set S such that $|{\bigcap }_{v\in E}{R}_v|=q\left(E\right)$ for each E ∈ . A necessary condition for the existence...