Irrepresentability by multiple intersection, or why the interval number is unbounded

AbstractWe consider the following question: Given a family of sets, is there a positive integer, t, so that every graph is the intersection graph of sets each of which is the union of t sets from the given family? We show that the answer is ‘no’ precisely when some bipartite graph fails to be the intersection graph of sets from the given family. We are especially interested in the case where the given family of sets generalizes the family of real intervals. We extend our results to uniform hypergraphs and simplicial complexes