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In combinatorics, a Sperner family (or Sperner system; named in honor of Emanuel Sperner), or clutter, is a family F of subsets of a finite set E in which none of the sets contains another. Equivalently, a Sperner family is an antichain in the inclusion lattice over the power set of E. A Sperner family is also sometimes called an independent system or irredundant set. Sperner families are counted by the Dedekind numbers, and their size is bounded by Sperner's theorem and the Lubell–Yamamoto–Meshalkin inequality. They may also be described in the language of hypergraphs rather than set families, where they are called clutters.

The k element subsets of an n element set form a Sperner family, the size of which is maximized when k = n/2 (or the nearest integer to it). Sperner's theorem states that these families are the largest possible Sperner families over an n element set. Formally, the theorem states that, for every Sperner family S over an n element set,

A clutter is a family of subsets of a finite set such that none contains any other; that is, it is a Sperner family. The difference is in the questions typically asked. Clutters are an important structure in the study of combinatorial optimization. (In more complicated language, a clutter is a hypergraph with the added property that whenever and (i. e. no edge properly contains another. An 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 .

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.

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.