Sperner family

In combinatorics, a Sperner family (or Sperner system), named in honor of Emanuel Sperner, is a family of sets (F, E) in which none of the sets is contained in 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 a clutter.

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 number of different Sperner families on a set of n elements is counted by the Dedekind numbers, the first few of which are

Although accurate asymptotic estimates are known for larger values of n, it is unknown whether there exists an exact formula that can be used to compute these numbers efficiently.

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,

The Lubell–Yamamoto–Meshalkin inequality provides another bound on the size of a Sperner family, and can be used to prove Sperner's theorem. It states that, if a_k denotes the number of sets of size k in a Sperner family over a set of n elements, then

A clutter H is a hypergraph ${\displaystyle (V,E)}$, with the added property that ${\displaystyle A\not \subseteq B}$ whenever ${\displaystyle A,B\in E}$ and ${\displaystyle A\neq B}$ (i.e. no edge properly contains another). That is, the sets of vertices represented by the hyperedges form a Sperner family. Clutters are an important structure in the study of combinatorial optimization. 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 E).

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