opposite notion to a clutter is an abstract simplicial complex, where every subset of an edge is contained in the hypergraph; this is an order ideal in the poset of subsets of V.) If is a clutter, then the blocker of H, denoted by , is the clutter with vertex set V and edge set consisting of all minimal sets so that for every . It can be shown that , so blockers give us a type of duality. We define to be the size of the largest collection of disjoint edges in H and to be the size of the smallest edge in . It is easy to see that . Examples If G is a simple loopless graph, then is a clutter (if edges are treated as unordered pairs of vertices) and is the collection of all minimal vertex covers. Here is the size of the largest matching and is the size of the smallest vertex cover. Kőnig's theorem states that, for bipartite graphs, . However for other graphs these two quantities may differ. Let G be a graph and let . The collection H of all edge-sets of s-t paths is a clutter and is the collection of all minimal edge cuts which separate s and t. In this case is the maximum number of edge-disjoint s-t paths, and is the size of the smallest edge-cut separating s and t, so Menger's theorem (edge-connectivity version) asserts that . Let G be a connected graph and let H be the clutter on consisting of all edge sets of spanning trees of G. Then is the collection of all minimal edge cutsets in G. Minors There is a minor relation on clutters which is similar to the minor relation on graphs. If is a clutter and , then we may delete v to get the clutter with vertex set and edge set consisting of all which do not contain v. We contract v to get the clutter . These two operations commute, and if J is another clutter, we say that J is a minor of H if a clutter isomorphic to J may be obtained from H by a sequence of deletions and contractions. References . . . . Families of sets