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| 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 (order)|lattice]] over the [[power set]] of ''E''. A Sperner family is also sometimes called an '''independent system''' or a '''clutter'''.
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| Sperner families are counted by the [[Dedekind number]]s, and their size is bounded by [[Sperner's theorem]] and the [[Lubell–Yamamoto–Meshalkin inequality]]. They may also be described in the language of [[hypergraph]]s rather than set families, where they are called '''clutters'''.
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| ==Dedekind numbers==
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| {{main|Dedekind number}}
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| The number of different Sperner families on a set of ''n'' elements is counted by the [[Dedekind number]]s, the first few of which are
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| :2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 {{OEIS|id=A000372}}. | |
| Although accurate [[asymptotic expansion|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.
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| ==Bounds on the size of a Sperner family== | |
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| === Sperner's theorem === | |
| {{main|Sperner's theorem}}
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| 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).
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| [[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,
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| :<math>|S| \le \binom{n}{\lfloor n/2\rfloor}.</math>
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| ===LYM inequality===
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| {{main|Lubell–Yamamoto–Meshalkin inequality}}
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| The [[Lubell–Yamamoto–Meshalkin inequality]] provides another bound on the size of a Sperner family, and can be used to prove Sperner's theorem.
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| It states that, if ''a<sub>k</sub>'' denotes the number of sets of size ''k'' in a Sperner family over a set of ''n'' elements, then
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| : <math>\sum_{k=0}^n\frac{a_k}{{n \choose k}} \le 1.</math>
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| ==Clutters==
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| A '''clutter''' ''H'' is a [[hypergraph]] <math>(V,E)</math>, with the added property that <math>A \not\subseteq B</math> whenever <math>A,B \in E</math> and <math>A \neq B</math> (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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| If <math>H = (V,E)</math> is a clutter, then the '''blocker''' of ''H'', denoted <math>b(H)</math>, is the clutter with vertex set ''V'' and edge set consisting of all minimal sets <math>B \subseteq V</math> so that <math>B \cap A \neq \varnothing</math> for every <math>A \in E</math>. It can be shown that <math>b(b(H)) = H</math> {{harv|Edmonds|Fulkerson|1970}}, so blockers give us a type of duality. We define <math>\nu(H)</math> to be the size of the largest collection of disjoint edges in ''H'' and <math>\tau(H)</math> to be the size of the smallest edge in <math>b(H)</math>. It is easy to see that <math>\nu(H) \le \tau(H)</math>.
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| === Examples ===
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| # If ''G'' is a simple loopless graph, then <math>H = (V(G),E(G))</math> is a clutter and <math>b(H)</math> is the collection of all minimal [[vertex cover]]s. Here <math>\nu(H)</math> is the size of the largest matching and <math>\tau(H)</math> is the size of the smallest vertex cover. [[König's theorem (graph theory)|König's theorem]] states that, for [[bipartite graph]]s, <math>\nu(H) = \tau(H)</math>. However for other graphs these two quantities may differ.
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| # Let ''G'' be a graph and let <math>s,t \in V(G)</math>. Define <math>H = (V,E)</math> by setting <math>V = E(G)</math> and letting ''E'' be the collection of all edge-sets of ''s''-''t'' paths. Then ''H'' is a clutter, and <math>b(H)</math> is the collection of all minimal edge cuts which separate ''s'' and ''t''. In this case <math>\nu(H)</math> is the maximum number of edge-disjoint ''s''-''t'' paths, and <math>\tau(H)</math> is the size of the smallest edge-cut separating ''s'' and ''t'', so [[Menger's theorem]] (edge-connectivity version) asserts that <math>\nu(H) = \tau(H)</math>.
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| # Let ''G'' be a connected graph and let ''H'' be the clutter on <math>E(G)</math> consisting of all edge sets of spanning trees of ''G''. Then <math>b(H)</math> is the collection of all minimal edge cuts in ''G''.
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| === Minors ===
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| There is a minor relation on clutters which is similar to the [[minor (graph theory)|minor relation]] on graphs. If <math>H = (V,E)</math> is a clutter and <math>v \in V</math>, then we may '''delete''' ''v'' to get the clutter <math>H \setminus v</math> with vertex set <math>
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| V \setminus \{v\}</math> and edge set consisting of all <math>A \in E</math> which do not contain ''v''. We '''contract''' ''v'' to get the clutter <math>H / v = b(b(H) \setminus v)</math>. 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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| == References ==
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| *{{citation
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| | last = Anderson | first = Ian
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| | title = Combinatorics of Finite Sets
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| | publisher = Oxford University Press
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| | year = 1987
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| | contribution = Sperner's theorem
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| | pages = 2–4}}.
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| *{{citation
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| | doi = 10.1016/S0021-9800(70)80083-7
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| | title = Bottleneck extrema
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| | last1 = Edmonds | first1 = J. | author1-link = Jack Edmonds
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| | last2 = Fulkerson | first2 = D. R. | author2-link = D. R. Fulkerson
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| | journal = [[Journal of Combinatorial Theory]]
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| | volume = 8
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| | issue = 3
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| | pages = 299–306
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| | year = 1970}}.
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| *{{citation
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| | last = Knuth | first = Donald E. | author-link = Donald Knuth
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| | contribution = Draft of Section 7.2.1.6: Generating All Trees
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| | title = [[The Art of Computer Programming]]
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| | volume = IV
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| | url = http://www-cs-faculty.stanford.edu/~knuth/fasc4a.ps.gz
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| | pages = 17–19
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| | year = 2005}}.
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| *{{citation
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| | last = Sperner | first = Emanuel | authorlink = Emanuel Sperner
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| | title = Ein Satz über Untermengen einer endlichen Menge
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| | journal = [[Mathematische Zeitschrift]]
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| | volume = 27 | issue = 1 | year = 1928
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| | doi = 10.1007/BF01171114
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| |language = German
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| | pages = 544–548 |jfm=54.0090.06 }}.
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| [[Category:Set families]]
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