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.

Dedekind numbers

Main article: Dedekind number

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

2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 (sequence A000372 in OEIS).

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.

Bounds on the size of a Sperner family

Sperner's theorem

Main article: Sperner's theorem

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,

$|S| \le \binom{n}{\lfloor n/2\rfloor}.$

LYM inequality

Main article: Lubell–Yamamoto–Meshalkin inequality

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

$\sum_{k=0}^n\frac{a_k}{{n \choose k}} \le 1.$

Clutters

A clutter H is a hypergraph $(V,E)$, with the added property that $A \not\subseteq B$ whenever $A,B \in E$ and $A \neq B$ (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).

If $H = (V,E)$ is a clutter, then the blocker of H, denoted $b(H)$, is the clutter with vertex set V and edge set consisting of all minimal sets $B \subseteq V$ so that $B \cap A \neq \varnothing$ for every $A \in E$. It can be shown that $b(b(H)) = H$ (Edmonds & Fulkerson 1970), so blockers give us a type of duality. We define $\nu(H)$ to be the size of the largest collection of disjoint edges in H and $\tau(H)$ to be the size of the smallest edge in $b(H)$. It is easy to see that $\nu(H) \le \tau(H)$.

Examples

1. If G is a simple loopless graph, then $H = (V(G),E(G))$ is a clutter and $b(H)$ is the collection of all minimal vertex covers. Here $\nu(H)$ is the size of the largest matching and $\tau(H)$ is the size of the smallest vertex cover. König's theorem states that, for bipartite graphs, $\nu(H) = \tau(H)$. However for other graphs these two quantities may differ.
2. Let G be a graph and let $s,t \in V(G)$. Define $H = (V,E)$ by setting $V = E(G)$ and letting E be the collection of all edge-sets of s-t paths. Then H is a clutter, and $b(H)$ is the collection of all minimal edge cuts which separate s and t. In this case $\nu(H)$ is the maximum number of edge-disjoint s-t paths, and $\tau(H)$ is the size of the smallest edge-cut separating s and t, so Menger's theorem (edge-connectivity version) asserts that $\nu(H) = \tau(H)$.
3. Let G be a connected graph and let H be the clutter on $E(G)$ consisting of all edge sets of spanning trees of G. Then $b(H)$ is the collection of all minimal edge cuts in G.

Minors

There is a minor relation on clutters which is similar to the minor relation on graphs. If $H = (V,E)$ is a clutter and $v \in V$, then we may delete v to get the clutter $H \setminus v$ with vertex set $V \setminus \{v\}$ and edge set consisting of all $A \in E$ which do not contain v. We contract v to get the clutter $H / v = b(b(H) \setminus v)$. 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.

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